2014

• Paper 1, Section I, D

  #Analysis I

  2014

  Show that every sequence of real numbers contains a monotone subsequence.

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• Paper 1, Section I, $4 \mathbf{F}$

  #Analysis I

  2014

  Find the radius of convergence of the following power series: (i) $\sum_{n \geqslant 1} \frac{n !}{n^{n}} z^{n}$; (ii) $\sum_{n \geqslant 1} n^{n} z^{n !}$.

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• Paper 1, Section II, D

  #Analysis I

  2014

  (a) Show that for all $x \in \mathbb{R}$,

  $$\lim _{k \rightarrow \infty} 3^{k} \sin \left(x / 3^{k}\right)=x,$$

  stating carefully what properties of sin you are using.

  Show that the series $\sum_{n \geqslant 1} 2^{n} \sin \left(x / 3^{n}\right)$ converges absolutely for all $x \in \mathbb{R}$.

  (b) Let $\left(a_{n}\right)_{n \in \mathbb{N}}$ be a decreasing sequence of positive real numbers tending to zero. Show that for $\theta \in \mathbb{R}, \theta$ not a multiple of $2 \pi$, the series

  $$\sum_{n \geqslant 1} a_{n} e^{i n \theta}$$

  converges.

  Hence, or otherwise, show that $\sum_{n \geqslant 1} \frac{\sin (n \theta)}{n}$ converges for all $\theta \in \mathbb{R}$.

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• Paper 1, Section II, E

  #Analysis I

  2014

  (i) State the Mean Value Theorem. Use it to show that if $f:(a, b) \rightarrow \mathbb{R}$ is a differentiable function whose derivative is identically zero, then $f$ is constant.

  (ii) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function and $\alpha>0$ a real number such that for all $x, y \in \mathbb{R}$,

  $$|f(x)-f(y)| \leqslant|x-y|^{\alpha} .$$

  Show that $f$ is continuous. Show moreover that if $\alpha>1$ then $f$ is constant.

  (iii) Let $f:[a, b] \rightarrow \mathbb{R}$ be continuous, and differentiable on $(a, b)$. Assume also that the right derivative of $f$ at $a$ exists; that is, the limit

  $$\lim _{x \rightarrow a+} \frac{f(x)-f(a)}{x-a}$$

  exists. Show that for any $\epsilon>0$ there exists $x \in(a, b)$ satisfying

  $$\left|\frac{f(x)-f(a)}{x-a}-f^{\prime}(x)\right|<\epsilon .$$

  [You should not assume that $f^{\prime}$ is continuous.]

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• Paper 1, Section II, E

  #Analysis I

  2014

  (i) Prove Taylor's Theorem for a function $f: \mathbb{R} \rightarrow \mathbb{R}$ differentiable $n$ times, in the following form: for every $x \in \mathbb{R}$ there exists $\theta$ with $0<\theta<1$ such that

  $$f(x)=\sum_{k=0}^{n-1} \frac{f^{(k)}(0)}{k !} x^{k}+\frac{f^{(n)}(\theta x)}{n !} x^{n}$$

  [You may assume Rolle's Theorem and the Mean Value Theorem; other results should be proved.]

  (ii) The function $f: \mathbb{R} \rightarrow \mathbb{R}$ is twice differentiable, and satisfies the differential equation $f^{\prime \prime}-f=0$ with $f(0)=A, f^{\prime}(0)=B$. Show that $f$ is infinitely differentiable. Write down its Taylor series at the origin, and prove that it converges to $f$ at every point. Hence or otherwise show that for any $a, h \in \mathbb{R}$, the series

  $$\sum_{k=0}^{\infty} \frac{f^{(k)}(a)}{k !} h^{k}$$

  converges to $f(a+h)$.

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• Paper 1, Section II, F

  #Analysis I

  2014

  Define what it means for a function $f:[0,1] \rightarrow \mathbb{R}$ to be (Riemann) integrable. Prove that $f$ is integrable whenever it is

  (a) continuous,

  (b) monotonic.

  Let $\left\{q_{k}: k \in \mathbb{N}\right\}$ be an enumeration of all rational numbers in $[0,1)$. Define a function $f:[0,1] \rightarrow \mathbb{R}$ by $f(0)=0$,

  $$f(x)=\sum_{k \in Q(x)} 2^{-k}, \quad x \in(0,1]$$

  where

  $$Q(x)=\left\{k \in \mathbb{N}: q_{k} \in[0, x)\right\}$$

  Show that $f$ has a point of discontinuity in every interval $I \subset[0,1]$.

  Is $f$ integrable? [Justify your answer.]

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• Paper 2, Section I, B

  #Differential Equations

  2014

  The following equation arises in the theory of elastic beams:

  $$t^{4} \frac{d^{2} u}{d t^{2}}+\lambda^{2} u=0, \quad \lambda>0, t>0$$

  where $u(t)$ is a real valued function.

  By using the change of variables

  $$t=\frac{1}{\tau}, \quad u(t)=\frac{v(\tau)}{\tau},$$

  find the general solution of the above equation.

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• Paper 2, Section I, B

  #Differential Equations

  2014

  Consider the ordinary differential equation

  $$P(x, y)+Q(x, y) \frac{d y}{d x}=0 .$$

  State an equation to be satisfied by $P$ and $Q$ that ensures that equation $(*)$ is exact. In this case, express the general solution of equation $(*)$ in terms of a function $F(x, y)$ which should be defined in terms of $P$ and $Q$.

  Consider the equation

  $$\frac{d y}{d x}=-\frac{4 x+3 y}{3 x+3 y^{2}}$$

  satisfying the boundary condition $y(1)=2$. Find an explicit relation between $y$ and $x$.

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• Paper 2, Section II, B

  #Differential Equations

  2014

  Use the transformation

  $$y(t)=\frac{1}{c x(t)} \frac{d x(t)}{d t}$$

  where $c$ is a constant, to map the Ricatti equation

  $$\frac{d y}{d t}+c y^{2}+a(t) y+b(t)=0, \quad t>0$$

  to a linear equation.

  Using the above result, as well as the change of variables $\tau=\ln t$, solve the boundary value problem

  $$\begin{gathered} \frac{d y}{d t}+y^{2}+\frac{y}{t}-\frac{\lambda^{2}}{t^{2}}=0, \quad t>0 \\ y(1)=2 \lambda \end{gathered}$$

  where $\lambda$ is a positive constant. What is the value of $t>0$ for which the solution is singular?

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• Paper 2, Section II, B

  #Differential Equations

  2014

  The so-called "shallow water theory" is characterised by the equations

  $$\begin{aligned} &\frac{\partial \zeta}{\partial t}+\frac{\partial}{\partial x}[(h+\zeta) u]=0 \\ &\frac{\partial u}{\partial t}+u \frac{\partial u}{\partial x}+g \frac{\partial \zeta}{\partial x}=0 \end{aligned}$$

  where $g$ denotes the gravitational constant, the constant $h$ denotes the undisturbed depth of the water, $u(x, t)$ denotes the speed in the $x$-direction, and $\zeta(x, t)$ denotes the elevation of the water.

  (i) Assuming that $|u|$ and $|\zeta|$ and their gradients are small in some appropriate dimensional considerations, show that $\zeta$ satisfies the wave equation

  $$\frac{\partial^{2} \zeta}{\partial t^{2}}=c^{2} \frac{\partial^{2} \zeta}{\partial x^{2}},$$

  where the constant $c$ should be determined in terms of $h$ and $g$.

  (ii) Using the change of variables

  $$\xi=x+c t, \quad \eta=x-c t$$

  show that the general solution of $(*)$ satisfying the initial conditions

  $$\zeta(x, 0)=u_{0}(x), \quad \frac{\partial \zeta}{\partial t}(x, 0)=v_{0}(x)$$

  is given by

  $$\zeta(x, t)=f(x+c t)+g(x-c t)$$

  where

  $$\begin{aligned} &\frac{d f(x)}{d x}=\frac{1}{2}\left[\frac{d u_{0}(x)}{d x}+\frac{1}{c} v_{0}(x)\right] \\ &\frac{d g(x)}{d x}=\frac{1}{2}\left[\frac{d u_{0}(x)}{d x}-\frac{1}{c} v_{0}(x)\right] \end{aligned}$$

  Simplify the above to find $\zeta$ in terms of $u_{0}$ and $v_{0}$.

  (iii) Find $\zeta(x, t)$ in the particular case that

  $$u_{0}(x)=H(x+1)-H(x-1), \quad v_{0}(x)=0, \quad-\infty<x<\infty$$

  where $H(\cdot)$ denotes the Heaviside step function.

  Describe in words this solution.

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• Paper 2, Section II, B

  #Differential Equations

  2014

  (a) Let $y_{1}(x)$ be a solution of the equation

  $$\frac{d^{2} y}{d x^{2}}+p(x) \frac{d y}{d x}+q(x) y=0$$

  Assuming that the second linearly independent solution takes the form $y_{2}(x)=$ $v(x) y_{1}(x)$, derive an ordinary differential equation for $v(x)$.

  (b) Consider the equation

  $$\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-2 x \frac{d y}{d x}+2 y=0, \quad-1<x<1 .$$

  By inspection or otherwise, find an explicit solution of this equation. Use the result in (a) to find the solution $y(x)$ satisfying the conditions

  $$y(0)=\frac{d y}{d x}(0)=1$$

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• Paper 2, Section II, B

  #Differential Equations

  2014

  Consider the damped pendulum equation

  $$\frac{d^{2} \theta}{d t^{2}}+c \frac{d \theta}{d t}+\sin \theta=0$$

  where $c$ is a positive constant. The energy $E$, which is the sum of the kinetic energy and the potential energy, is defined by

  $$E(t)=\frac{1}{2}\left(\frac{d \theta}{d t}\right)^{2}+1-\cos \theta$$

  (i) Verify that $E(t)$ is a decreasing function.

  (ii) Assuming that $\theta$ is sufficiently small, so that terms of order $\theta^{3}$ can be neglected, find an approximation for the general solution of $(*)$ in terms of two arbitrary constants. Discuss the dependence of this approximate solution on $c$.

  (iii) By rewriting $(*)$ as a system of equations for $x(t)=\theta$ and $y(t)=\dot{\theta}$, find all stationary points of $(*)$ and discuss their nature for all $c$, except $c=2$.

  (iv) Draw the phase plane curves for the particular case $c=1$.

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• Paper 4, Section I, $3 C$

  #Dynamics and Relativity

  2014

  A particle of mass $m$ has charge $q$ and moves in a constant magnetic field B. Show that the particle's path describes a helix. In which direction is the axis of the helix, and what is the particle's rotational angular frequency about that axis?

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• Paper 4, Section I, $4 \mathrm{C}$

  #Dynamics and Relativity

  2014

  What is a 4-vector? Define the inner product of two 4-vectors and give the meanings of the terms timelike, null and spacelike. How do the four components of a 4-vector change under a Lorentz transformation of speed $v$ ? [Without loss of generality, you may take the velocity of the transformation to be along the positive $x$-axis.]

  Show that a 4-vector that is timelike in one frame of reference is also timelike in a second frame of reference related by a Lorentz transformation. [Again, you may without loss of generality take the velocity of the transformation to be along the positive $x$-axis.]

  Show that any null 4-vector may be written in the form $a(1, \hat{\mathbf{n}})$ where $a$ is real and $\hat{\mathbf{n}}$ is a unit 3-vector. Given any two null 4-vectors that are future-pointing, that is, which have positive time-components, show that their sum is either null or timelike.

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• Paper 4, Section II, C

  #Dynamics and Relativity

  2014

  A rocket of mass $m(t)$, which includes the mass of its fuel and everything on board, moves through free space in a straight line at speed $v(t)$. When its engines are operational, they burn fuel at a constant mass rate $\alpha$ and eject the waste gases behind the rocket at a constant speed $u$ relative to the rocket. Obtain the rocket equation

  $$m \frac{d v}{d t}-\alpha u=0$$

  The rocket is initially at rest in a cloud of space dust which is also at rest. The engines are started and, as the rocket travels through the cloud, it collects dust which it stores on board for research purposes. The mass of dust collected in a time $\delta t$ is given by $\beta \delta x$, where $\delta x$ is the distance travelled in that time and $\beta$ is a constant. Obtain the new equations

  $$\begin{aligned} \frac{d m}{d t} &=\beta v-\alpha \\ m \frac{d v}{d t} &=\alpha u-\beta v^{2} \end{aligned}$$

  By eliminating $t$, or otherwise, obtain the relationship

  $$m=\lambda m_{0} u \sqrt{\frac{(\lambda u-v)^{\lambda-1}}{(\lambda u+v)^{\lambda+1}}},$$

  where $m_{0}$ is the initial mass of the rocket and $\lambda=\sqrt{\alpha / \beta u}$.

  If $\lambda>1$, show that the fuel will be exhausted before the speed of the rocket can reach $\lambda u$. Comment on the case when $\lambda<1$, giving a physical interpretation of your answer.

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• Paper 4, Section II, C

  #Dynamics and Relativity

  2014

  A reference frame $S^{\prime}$ rotates with constant angular velocity $\boldsymbol{\omega}$ relative to an inertial frame $S$ that has the same origin as $S^{\prime}$. A particle of mass $m$ at position vector $\mathbf{x}$ is subject to a force $\mathbf{F}$. Derive the equation of motion for the particle in $S^{\prime}$.

  A marble moves on a smooth plane which is inclined at an angle $\theta$ to the horizontal. The whole plane rotates at constant angular speed $\omega$ about a vertical axis through a point $O$ fixed in the plane. Coordinates $(\xi, \eta)$ are defined with respect to axes fixed in the plane: $O \xi$ horizontal and $O \eta$ up the line of greatest slope in the plane. Ensuring that you account for the normal reaction force, show that the motion of the marble obeys

  $$\begin{aligned} \ddot{\xi} &=\omega^{2} \xi+2 \omega \dot{\eta} \cos \theta, \\ \ddot{\eta} &=\omega^{2} \eta \cos ^{2} \theta-2 \omega \dot{\xi} \cos \theta-g \sin \theta \end{aligned}$$

  By considering the marble's kinetic energy as measured on the plane in the rotating frame, or otherwise, find a constant of the motion.

  [You may assume that the marble never leaves the plane.]

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• Paper 4, Section II, C

  #Dynamics and Relativity

  2014

  A thin flat disc of radius $a$ has density (mass per unit area) $\rho(r, \theta)=\rho_{0}(a-r)$ where $(r, \theta)$ are plane polar coordinates on the disc and $\rho_{0}$ is a constant. The disc is free to rotate about a light, thin rod that is rigidly fixed in space, passing through the centre of the disc orthogonal to it. Find the moment of inertia of the disc about the rod.

  The section of the disc lying in $r \geqslant \frac{1}{2} a,-\frac{\pi}{13} \leqslant \theta \leqslant \frac{\pi}{13}$ is cut out and removed. Starting from rest, a constant torque $\tau$ is applied to the remaining part of the disc until its angular speed about the axis reaches $\Omega$. Show that this takes a time

  $$\frac{3 \pi \rho_{0} a^{5} \Omega}{32 \tau}$$

  After this time, no further torque is applied and the partial disc continues to rotate at constant angular speed $\Omega$. Given that the total mass of the partial disc is $k \rho_{0} a^{3}$, where $k$ is a constant that you need not determine, find the position of the centre of mass, and hence its acceleration. From where does the force required to produce this acceleration arise?

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• Paper 4, Section II, C

  #Dynamics and Relativity

  2014

  Define the 4-momentum of a particle and describe briefly the principle of conservation of 4-momentum.

  A photon of angular frequency $\omega$ is absorbed by a particle of rest mass $m$ that is stationary in the laboratory frame of reference. The particle then splits into two equal particles, each of rest mass $\mathrm{\alpha m}$.

  Find the maximum possible value of $\alpha$ as a function of $\mu=\hbar \omega / m c^{2}$. Verify that as $\mu \rightarrow 0$, this maximum value tends to $\frac{1}{2}$. For general $\mu$, show that when the maximum value of $\alpha$ is achieved, the resulting particles are each travelling at speed $c /\left(1+\mu^{-1}\right)$ in the laboratory frame.

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• Paper 3, Section I, D

  #Groups

  2014

  Let $G=\mathbb{Q}$ be the rational numbers, with addition as the group operation. Let $x, y$ be non-zero elements of $G$, and let $N \leqslant G$ be the subgroup they generate. Show that $N$ is isomorphic to $\mathbb{Z}$.

  Find non-zero elements $x, y \in \mathbb{R}$ which generate a subgroup that is not isomorphic to $\mathbb{Z}$.

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• Paper 3, Section I, D

  #Groups

  2014

  Let $G$ be a group, and suppose the centre of $G$ is trivial. If $p$ divides $|G|$, show that $G$ has a non-trivial conjugacy class whose order is prime to $p$.

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• Paper 3, Section II, D

  #Groups

  2014

  Let $S_{n}$ be the group of permutations of $\{1, \ldots, n\}$, and suppose $n$ is even, $n \geqslant 4$.

  Let $g=(12) \in S_{n}$, and $h=(12)(34) \ldots(n-1 n) \in S_{n}$.

  (i) Compute the centraliser of $g$, and the orders of the centraliser of $g$ and of the centraliser of $h$.

  (ii) Now let $n=6$. Let $G$ be the group of all symmetries of the cube, and $X$ the set of faces of the cube. Show that the action of $G$ on $X$ makes $G$ isomorphic to the centraliser of $h$ in $S_{6}$. [Hint: Show that $-1 \in G$ permutes the faces of the cube according to $h$.]

  Show that $G$ is also isomorphic to the centraliser of $g$ in $S_{6}$.

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• Paper 3, Section II, D

  #Groups

  2014

  Let $p$ be a prime number. Let $G$ be a group such that every non-identity element of $G$ has order $p$.

  (i) Show that if $|G|$ is finite, then $|G|=p^{n}$ for some $n$. [You must prove any theorems that you use.]

  (ii) Show that if $H \leqslant G$, and $x \notin H$, then $\langle x\rangle \cap H=\{1\}$.

  Hence show that if $G$ is abelian, and $|G|$ is finite, then $G \simeq C_{p} \times \cdots \times C_{p}$.

  (iii) Let $G$ be the set of all $3 \times 3$ matrices of the form

  $$\left(\begin{array}{lll} 1 & a & x \\ 0 & 1 & b \\ 0 & 0 & 1 \end{array}\right)$$

  where $a, b, x \in \mathbb{F}_{p}$ and $\mathbb{F}_{p}$ is the field of integers modulo $p$. Show that every nonidentity element of $G$ has order $p$ if and only if $p>2$. [You may assume that $G$ is a subgroup of the group of all $3 \times 3$ invertible matrices.]

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• Paper 3, Section II, D

  #Groups

  2014

  Let $p$ be a prime number, and $G=G L_{2}\left(\mathbb{F}_{p}\right)$, the group of $2 \times 2$ invertible matrices with entries in the field $\mathbb{F}_{p}$ of integers modulo $p$.

  The group $G$ acts on $X=\mathbb{F}_{p} \cup\{\infty\}$ by Möbius transformations,

  $$\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) \cdot z=\frac{a z+b}{c z+d}$$

  (i) Show that given any distinct $x, y, z \in X$ there exists $g \in G$ such that $g \cdot 0=x$, $g \cdot 1=y$ and $g \cdot \infty=z$. How many such $g$ are there?

  (ii) $G$ acts on $X \times X \times X$ by $g \cdot(x, y, z)=(g \cdot x, g \cdot y, g \cdot z)$. Describe the orbits, and for each orbit, determine its stabiliser, and the orders of the orbit and stabiliser.

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• Paper 3, Section II, D

  #Groups

  2014

  (a) Let $G$ be a group, and $N$ a subgroup of $G$. Define what it means for $N$ to be normal in $G$, and show that if $N$ is normal then $G / N$ naturally has the structure of a group.

  (b) For each of (i)-(iii) below, give an example of a non-trivial finite group $G$ and non-trivial normal subgroup $N \leqslant G$ satisfying the stated properties.

  (i) $G / N \times N \simeq G$.

  (ii) There is no group homomorphism $G / N \rightarrow G$ such that the composite $G / N \rightarrow G \rightarrow G / N$ is the identity.

  (iii) There is a group homomorphism $i: G / N \rightarrow G$ such that the composite $G / N \rightarrow G \rightarrow G / N$ is the identity, but the map

  $$G / N \times N \rightarrow G, \quad(g N, n) \mapsto i(g N) n$$

  is not a group homomorphism.

  Show also that for any $N \leqslant G$ satisfying (iii), this map is always a bijection.

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• Paper 4, Section I, E

  #Numbers and Sets

  2014

  Use Euclid's algorithm to determine $d$, the greatest common divisor of 203 and 147 , and to express it in the form $203 x+147 y$ for integers $x, y$. Hence find all solutions in integers $x, y$ of the equation $203 x+147 y=d$.

  How many integers $n$ are there with $1 \leqslant n \leqslant 2014$ and $21 n \equiv 25(\bmod 29) ?$

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• Paper 4, Section I, $2 \mathrm{E}$

  #Numbers and Sets

  2014

  Define the binomial coefficients $\left(\begin{array}{l}n \\ k\end{array}\right)$, for integers $n, k$ satisfying $n \geqslant k \geqslant 0$. Prove directly from your definition that if $n>k \geqslant 0$ then

  $$\left(\begin{array}{l} n \\ k \end{array}\right)+\left(\begin{array}{c} n \\ k+1 \end{array}\right)=\left(\begin{array}{c} n+1 \\ k+1 \end{array}\right)$$

  and that for every $m \geqslant 0$ and $n \geqslant 0$,

  $$\sum_{k=0}^{m}\left(\begin{array}{c} n+k \\ k \end{array}\right)=\left(\begin{array}{c} n+m+1 \\ m \end{array}\right)$$

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• Paper 4, Section II, E

  #Numbers and Sets

  2014

  What does it mean to say that the sequence of real numbers $\left(x_{n}\right)$ converges to the limit $x ?$ What does it mean to say that the series $\sum_{n=1}^{\infty} x_{n}$ converges to $s$ ?

  Let $\sum_{n=1}^{\infty} a_{n}$ and $\sum_{n=1}^{\infty} b_{n}$ be convergent series of positive real numbers. Suppose that $\left(x_{n}\right)$ is a sequence of positive real numbers such that for every $n \geqslant 1$, either $x_{n} \leqslant a_{n}$ or $x_{n} \leqslant b_{n}$. Show that $\sum_{n=1}^{\infty} x_{n}$ is convergent.

  Show that $\sum_{n=1}^{\infty} 1 / n^{2}$ is convergent, and that $\sum_{n=1}^{\infty} 1 / n^{\alpha}$ is divergent if $\alpha \leqslant 1$.

  Let $\left(x_{n}\right)$ be a sequence of positive real numbers such that $\sum_{n=1}^{\infty} n^{2} x_{n}^{2}$ is convergent. Show that $\sum_{n=1}^{\infty} x_{n}$ is convergent. Determine (with proof or counterexample) whether or not the converse statement holds.

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• Paper 4, Section II, E

  #Numbers and Sets

  2014

  (i) State and prove the Fermat-Euler Theorem.

  (ii) Let $p$ be an odd prime number, and $x$ an integer coprime to $p$. Show that $x^{(p-1) / 2} \equiv \pm 1(\bmod p)$, and that if the congruence $y^{2} \equiv x(\bmod p)$ has a solution then $x^{(p-1) / 2} \equiv 1(\bmod p)$.

  (iii) By arranging the residue classes coprime to $p$ into pairs $\{a, b x\}$ with $a b \equiv 1(\bmod p)$, or otherwise, show that if the congruence $y^{2} \equiv x(\bmod p)$ has no solution then $x^{(p-1) / 2} \equiv-1(\bmod p) .$

  (iv) Show that $5^{5^{5}} \equiv 5(\bmod 23)$.

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• Paper 4, Section II, E

  #Numbers and Sets

  2014

  (i) What does it mean to say that a set $X$ is countable? Show directly that the set of sequences $\left(x_{n}\right)_{n \in \mathbb{N}}$, with $x_{n} \in\{0,1\}$ for all $n$, is uncountable.

  (ii) Let $S$ be any subset of $\mathbb{N}$. Show that there exists a bijection $f: \mathbb{N} \rightarrow \mathbb{N}$ such that $f(S)=2 \mathbb{N}$ (the set of even natural numbers) if and only if both $S$ and its complement are infinite.

  (iii) Let $\sqrt{2}=1 \cdot a_{1} a_{2} a_{3} \ldots$ be the binary expansion of $\sqrt{2}$. Let $X$ be the set of all sequences $\left(x_{n}\right)$ with $x_{n} \in\{0,1\}$ such that for infinitely many $n, x_{n}=0$. Let $Y$ be the set of all $\left(x_{n}\right) \in X$ such that for infinitely many $n, x_{n}=a_{n}$. Show that $Y$ is uncountable.

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• Paper 4, Section II, E

  #Numbers and Sets

  2014

  (i) State and prove the Inclusion-Exclusion Principle.

  (ii) Let $n>1$ be an integer. Denote by $\mathbb{Z} / n \mathbb{Z}$ the integers modulo $n$. Let $X$ be the set of all functions $f: \mathbb{Z} / n \mathbb{Z} \rightarrow \mathbb{Z} / n \mathbb{Z}$ such that for every $j \in \mathbb{Z} / n \mathbb{Z}, f(j)-f(j-1) \not \equiv j$ $(\bmod n)$. Show that

  $$|X|= \begin{cases}(n-1)^{n}+1-n & \text { if } n \text { is odd } \\ (n-1)^{n}-1 & \text { if } n \text { is even }\end{cases}$$

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• Paper 2, Section I, F

  #Probability

  2014

  Consider a particle situated at the origin $(0,0)$ of $\mathbb{R}^{2}$. At successive times a direction is chosen independently by picking an angle uniformly at random in the interval $[0,2 \pi]$, and the particle then moves an Euclidean unit length in this direction. Find the expected squared Euclidean distance of the particle from the origin after $n$ such movements.

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• Paper 2, Section I, F

  #Probability

  2014

  Consider independent discrete random variables $X_{1}, \ldots, X_{n}$ and assume $E\left[X_{i}\right]$ exists for all $i=1, \ldots, n$.

  Show that

  $$E\left[\prod_{i=1}^{n} X_{i}\right]=\prod_{i=1}^{n} E\left[X_{i}\right]$$

  If the $X_{1}, \ldots, X_{n}$ are also positive, show that

  $$\prod_{i=1}^{n} \sum_{m=0}^{\infty} P\left(X_{i}>m\right)=\sum_{m=0}^{\infty} P\left(\prod_{i=1}^{n} X_{i}>m\right)$$

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• Paper 2, Section II, 9F

  #Probability

  2014

  State the axioms of probability.

  State and prove Boole's inequality.

  Suppose you toss a sequence of coins, the $i$-th of which comes up heads with probability $p_{i}$, where $\sum_{i=1}^{\infty} p_{i}<\infty$. Calculate the probability of the event that infinitely many heads occur.

  Suppose you repeatedly and independently roll a pair of fair dice and each time record the sum of the dice. What is the probability that an outcome of 5 appears before an outcome of 7 ? Justify your answer.

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• Paper 2, Section II, F

  #Probability

  2014

  Define what it means for a random variable $X$ to have a Poisson distribution, and find its moment generating function.

  Suppose $X, Y$ are independent Poisson random variables with parameters $\lambda, \mu$. Find the distribution of $X+Y$.

  If $X_{1}, \ldots, X_{n}$ are independent Poisson random variables with parameter $\lambda=1$, find the distribution of $\sum_{i=1}^{n} X_{i}$. Hence or otherwise, find the limit of the real sequence

  $$a_{n}=e^{-n} \sum_{j=0}^{n} \frac{n^{j}}{j !}, \quad n \in \mathbb{N}$$

  [Standard results may be used without proof provided they are clearly stated.]

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• Paper 2, Section II, F

  #Probability

  2014

  For any function $g: \mathbb{R} \rightarrow \mathbb{R}$ and random variables $X, Y$, the "tower property" of conditional expectations is

  $$E[g(X)]=E[E[g(X) \mid Y]] .$$

  Provide a proof of this property when both $X, Y$ are discrete.

  Let $U_{1}, U_{2}, \ldots$ be a sequence of independent uniform $U(0,1)$-random variables. For $x \in[0,1]$ find the expected number of $U_{i}$ 's needed such that their sum exceeds $x$, that is, find $E[N(x)]$ where

  $$N(x)=\min \left\{n: \sum_{i=1}^{n} U_{i}>x\right\}$$

  [Hint: Write $\left.E[N(x)]=E\left[E\left[N(x) \mid U_{1}\right]\right] .\right]$

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• Paper 2, Section II, F

  #Probability

  2014

  Give the definition of an exponential random variable $X$ with parameter $\lambda$. Show that $X$ is memoryless.

  Now let $X, Y$ be independent exponential random variables, each with parameter $\lambda$. Find the probability density function of the random variable $Z=\min (X, Y)$ and the probability $P(X>Y)$.

  Suppose the random variables $G_{1}, G_{2}$ are independent and each has probability density function given by

  $$f(y)=C^{-1} e^{-y} y^{-1 / 2}, \quad y>0, \quad \text { where } C=\int_{0}^{\infty} e^{-y} y^{-1 / 2} d y$$

  Find the probability density function of $G_{1}+G_{2} \cdot$ [You may use standard results without proof provided they are clearly stated.]

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• Paper 3, Section I, A

  #Vector Calculus

  2014

  (a) For $\mathbf{x} \in \mathbb{R}^{n}$ and $r=|\mathbf{x}|$, show that

  $$\frac{\partial r}{\partial x_{i}}=\frac{x_{i}}{r}$$

  (b) Use index notation and your result in (a), or otherwise, to compute

  (i) $\nabla \cdot(f(r) \mathbf{x})$, and

  (ii) $\nabla \times(f(r) \mathbf{x})$ for $n=3$.

  (c) Show that for each $n \in \mathbb{N}$ there is, up to an arbitrary constant, just one vector field $\mathbf{F}(\mathbf{x})$ of the form $f(r) \mathbf{x}$ such that $\nabla \cdot \mathbf{F}(\mathbf{x})=0$ everywhere on $\mathbb{R}^{n} \backslash\{\mathbf{0}\}$, and determine $\mathbf{F}$.

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• Paper 3, Section I, A

  #Vector Calculus

  2014

  Let $\mathbf{F}(\mathbf{x})$ be a vector field defined everywhere on the domain $G \subset \mathbb{R}^{3}$.

  (a) Suppose that $\mathbf{F}(\mathbf{x})$ has a potential $\phi(\mathbf{x})$ such that $\mathbf{F}(\mathbf{x})=\nabla \phi(\mathbf{x})$ for $\mathbf{x} \in G$. Show that

  $$\int_{\gamma} \mathbf{F} \cdot \mathbf{d} \mathbf{x}=\phi(\mathbf{b})-\phi(\mathbf{a})$$

  for any smooth path $\gamma$ from a to $\mathbf{b}$ in $G$. Show further that necessarily $\nabla \times \mathbf{F}=\mathbf{0}$ on $G$.

  (b) State a condition for $G$ which ensures that $\nabla \times \mathbf{F}=\mathbf{0}$ implies $\int_{\gamma} \mathbf{F} \cdot \mathbf{d x}$ is pathindependent.

  (c) Compute the line integral $\oint_{\gamma} \mathbf{F} \cdot \mathbf{d} \mathbf{x}$ for the vector field

  $$\mathbf{F}(\mathbf{x})=\left(\begin{array}{c} \frac{-y}{x^{2}+y^{2}} \\ \frac{x}{x^{2}+y^{2}} \\ 0 \end{array}\right)$$

  where $\gamma$ denotes the anti-clockwise path around the unit circle in the $(x, y)$-plane. Compute $\nabla \times \mathbf{F}$ and comment on your result in the light of (b).

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• Paper 3, Section II, A

  #Vector Calculus

  2014

  The surface $C$ in $\mathbb{R}^{3}$ is given by $z^{2}=x^{2}+y^{2}$.

  (a) Show that the vector field

  $$\mathbf{F}(\mathbf{x})=\left(\begin{array}{l} x \\ y \\ z \end{array}\right)$$

  is tangent to the surface $C$ everywhere.

  (b) Show that the surface integral $\int_{S} \mathbf{F} \cdot \mathbf{d} \mathbf{S}$ is a constant independent of $S$ for any surface $S$ which is a subset of $C$, and determine this constant.

  (c) The volume $V$ in $\mathbb{R}^{3}$ is bounded by the surface $C$ and by the cylinder $x^{2}+y^{2}=1$. Sketch $V$ and compute the volume integral

  $$\int_{V} \nabla \cdot \mathbf{F} d V$$

  directly by integrating over $V$.

  (d) Use the Divergence Theorem to verify the result you obtained in part (b) for the integral $\int_{S} \mathbf{F} \cdot \mathbf{d} \mathbf{S}$, where $S$ is the portion of $C$ lying in $-1 \leqslant z \leqslant 1$.

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• Paper 3, Section II, A

  #Vector Calculus

  2014

  (a) State Stokes' Theorem for a surface $S$ with boundary $\partial S$.

  (b) Let $S$ be the surface in $\mathbb{R}^{3}$ given by $z^{2}=1+x^{2}+y^{2}$ where $\sqrt{2} \leqslant z \leqslant \sqrt{5}$. Sketch the surface $S$ and find the surface element $\mathbf{d} \mathbf{S}$ with respect to the Cartesian coordinates $x$ and $y$.

  (c) Compute $\nabla \times \mathbf{F}$ for the vector field

  $$\mathbf{F}(\mathbf{x})=\left(\begin{array}{c} -y \\ x \\ x y(x+y) \end{array}\right)$$

  and verify Stokes' Theorem for $\mathbf{F}$ on the surface $S$.

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• Paper 3, Section II, 11A

  #Vector Calculus

  2014

  (i) Starting with Poisson's equation in $\mathbb{R}^{3}$,

  $$\nabla^{2} \phi(\mathbf{x})=f(\mathbf{x})$$

  derive Gauss' flux theorem

  $$\int_{V} f(\mathbf{x}) d V=\int_{\partial V} \mathbf{F}(\mathbf{x}) \cdot \mathbf{d} \mathbf{S}$$

  for $\mathbf{F}(\mathbf{x})=\nabla \phi(\mathbf{x})$ and for any volume $V \subseteq \mathbb{R}^{3}$.

  (ii) Let

  $$I=\int_{S} \frac{\mathbf{x} \cdot \mathbf{d} \mathbf{S}}{|\mathbf{x}|^{3}} .$$

  Show that $I=4 \pi$ if $S$ is the sphere $|\mathbf{x}|=R$, and that $I=0$ if $S$ bounds a volume that does not contain the origin.

  (iii) Show that the electric field defined by

  $$\mathbf{E}(\mathbf{x})=\frac{q}{4 \pi \epsilon_{0}} \frac{\mathbf{x}-\mathbf{a}}{|\mathbf{x}-\mathbf{a}|^{3}}, \quad \mathbf{x} \neq \mathbf{a}$$

  satisfies

  $$\int_{\partial V} \mathbf{E} \cdot \mathbf{d} \mathbf{S}= \begin{cases}0 & \text { if } \mathbf{a} \notin V \\ \frac{q}{\epsilon_{0}} & \text { if } \mathbf{a} \in V\end{cases}$$

  where $\partial V$ is a surface bounding a closed volume $V$ and $\mathbf{a} \notin \partial V$, and where the electric charge $q$ and permittivity of free space $\epsilon_{0}$ are constants. This is Gauss' law for a point electric charge.

  (iv) Assume that $f(\mathbf{x})$ is spherically symmetric around the origin, i.e., it is a function only of $|\mathbf{x}|$. Assume that $\mathbf{F}(\mathbf{x})$ is also spherically symmetric. Show that $\mathbf{F}(\mathbf{x})$ depends only on the values of $f$ inside the sphere with radius $|\mathbf{x}|$ but not on the values of $f$ outside this sphere.

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• Paper 3, Section II, A

  #Vector Calculus

  2014

  (a) Show that any rank 2 tensor $t_{i j}$ can be written uniquely as a sum of two rank 2 tensors $s_{i j}$ and $a_{i j}$ where $s_{i j}$ is symmetric and $a_{i j}$ is antisymmetric.

  (b) Assume that the rank 2 tensor $t_{i j}$ is invariant under any rotation about the $z$-axis, as well as under a rotation of angle $\pi$ about any axis in the $(x, y)$-plane through the origin.

  (i) Show that there exist $\alpha, \beta \in \mathbb{R}$ such that $t_{i j}$ can be written as

  $$t_{i j}=\alpha \delta_{i j}+\beta \delta_{i 3} \delta_{j 3} .$$

  (ii) Is there some proper subgroup of the rotations specified above for which the result $(*)$ still holds if the invariance of $t_{i j}$ is restricted to this subgroup? If so, specify the smallest such subgroup.

  (c) The array of numbers $d_{i j k}$ is such that $d_{i j k} s_{i j}$ is a vector for any symmetric matrix $s_{i j}$.

  (i) By writing $d_{i j k}$ as a sum of $d_{i j k}^{s}$ and $d_{i j k}^{a}$ with $d_{i j k}^{s}=d_{j i k}^{s}$ and $d_{i j k}^{a}=-d_{j i k}^{a}$, show that $d_{i j k}^{s}$ is a rank 3 tensor. [You may assume without proof the Quotient Theorem for tensors.]

  (ii) Does $d_{i j k}^{a}$ necessarily have to be a tensor? Justify your answer.

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• Paper 1, Section I, 1B

  #Vectors and Matrices

  2014

  (a) Let

  $$z=2+2 i$$

  (i) Compute $z^{4}$.

  (ii) Find all complex numbers $w$ such that $w^{4}=z$.

  (b) Find all the solutions of the equation

  $$e^{2 z^{2}}-1=0$$

  (c) Let $z=x+i y, \bar{z}=x-i y, x, y \in \mathbb{R}$. Show that the equation of any line, and of any circle, may be written respectively as

  $$B z+\bar{B} \bar{z}+C=0 \quad \text { and } \quad z \bar{z}+\bar{B} z+B \bar{z}+C=0$$

  for some complex $B$ and real $C$.

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• Paper 1, Section I, 2A

  #Vectors and Matrices

  2014

  (a) What is meant by an eigenvector and the corresponding eigenvalue of a matrix $A$ ?

  (b) Let $A$ be the matrix

  $$A=\left(\begin{array}{ccc} 3 & -2 & -2 \\ 1 & 0 & -2 \\ 3 & -3 & -1 \end{array}\right)$$

  Find the eigenvalues and the corresponding eigenspaces of $A$ and determine whether or not $A$ is diagonalisable.

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• Paper 1, Section II, 5B

  #Vectors and Matrices

  2014

  (i) For vectors $\mathbf{a}, \mathbf{b}, \mathbf{c} \in \mathbb{R}^{3}$, show that

  $$\mathbf{a} \times(\mathbf{b} \times \mathbf{c})=(\mathbf{a} \cdot \mathbf{c}) \mathbf{b}-(\mathbf{a} \cdot \mathbf{b}) \mathbf{c} .$$

  Show that the plane $(\mathbf{r}-\mathbf{a}) \cdot \mathbf{n}=0$ and the line $(\mathbf{r}-\mathbf{b}) \times \mathbf{m}=\mathbf{0}$, where $\mathbf{m} \cdot \mathbf{n} \neq 0$, intersect at the point

  $$\mathbf{r}=\frac{(\mathbf{a} \cdot \mathbf{n}) \mathbf{m}+\mathbf{n} \times(\mathbf{b} \times \mathbf{m})}{\mathbf{m} \cdot \mathbf{n}}$$

  and only at that point. What happens if $\mathbf{m} \cdot \mathbf{n}=0$ ?

  (ii) Explain why the distance between the planes $\left(\mathbf{r}-\mathbf{a}_{1}\right) \cdot \hat{\mathbf{n}}=0$ and $\left(\mathbf{r}-\mathbf{a}_{2}\right) \cdot \hat{\mathbf{n}}=0$ is $\left|\left(\mathbf{a}_{1}-\mathbf{a}_{2}\right) \cdot \hat{\mathbf{n}}\right|$, where $\hat{\mathbf{n}}$ is a unit vector.

  (iii) Find the shortest distance between the lines $(3+s, 3 s, 4-s)$ and $(-2,3+t, 3-t)$ where $s, t \in \mathbb{R}$. [You may wish to consider two appropriately chosen planes and use the result of part (ii).]

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• Paper 1, Section II, A

  #Vectors and Matrices

  2014

  Let $A$ be a real $n \times n$ symmetric matrix.

  (i) Show that all eigenvalues of $A$ are real, and that the eigenvectors of $A$ with respect to different eigenvalues are orthogonal. Assuming that any real symmetric matrix can be diagonalised, show that there exists an orthonormal basis $\left\{\mathbf{y}_{i}\right\}$ of eigenvectors of $A$.

  (ii) Consider the linear system

  $$A \mathbf{x}=\mathbf{b} .$$

  Show that this system has a solution if and only if $\mathbf{b} \cdot \mathbf{h}=0$ for every vector $\mathbf{h}$ in the kernel of $A$. Let $\mathbf{x}$ be such a solution. Given an eigenvector of $A$ with non-zero eigenvalue, determine the component of $x$ in the direction of this eigenvector. Use this result to find the general solution of the linear system, in the form

  $$\mathbf{x}=\sum_{i=1}^{n} \alpha_{i} \mathbf{y}_{i}$$

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• Paper 1, Section II, $7 \mathrm{C}$

  #Vectors and Matrices

  2014

  Let $\mathcal{A}: \mathbb{C}^{2} \rightarrow \mathbb{C}^{2}$ be the linear map

  $$\mathcal{A}\left(\begin{array}{c} z \\ w \end{array}\right)=\left(\begin{array}{c} z \mathrm{e}^{i \theta}+w \\ w \mathrm{e}^{-i \phi}+z \end{array}\right)$$

  where $\theta$ and $\phi$ are real constants. Write down the matrix $A$ of $\mathcal{A}$ with respect to the standard basis of $\mathbb{C}^{2}$ and show that $\operatorname{det} A=2 i \sin \frac{1}{2}(\theta-\phi) \exp \left(\frac{1}{2} i(\theta-\phi)\right)$.

  Let $\mathcal{R}: \mathbb{C}^{2} \rightarrow \mathbb{R}^{4}$ be the invertible map

  $$\mathcal{R}\left(\begin{array}{c} z \\ w \end{array}\right)=\left(\begin{array}{l} \operatorname{Re} z \\ \operatorname{Im} z \\ \operatorname{Re} w \\ \operatorname{Im} w \end{array}\right)$$

  and define a linear map $\mathcal{B}: \mathbb{R}^{4} \rightarrow \mathbb{R}^{4}$ by $\mathcal{B}=\mathcal{R} \mathcal{A} \mathcal{R}^{-1}$. Find the image of each of the standard basis vectors of $\mathbb{R}^{4}$ under both $\mathcal{R}^{-1}$ and $\mathcal{B}$. Hence, or otherwise, find the matrix $B$ of $\mathcal{B}$ with respect to the standard basis of $\mathbb{R}^{4}$ and verify that $\operatorname{det} B=|\operatorname{det} A|^{2}$.

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• Paper 3, Section I, F

  #Analysis II

  2014

  Let $U \subset \mathbb{R}^{n}$ be an open set and let $f: U \rightarrow \mathbb{R}$ be a differentiable function on $U$ such that $\left\|\left.D f\right|_{x}\right\| \leqslant M$ for some constant $M$ and all $x \in U$, where $\left\|\left.D f\right|_{x}\right\|$ denotes the operator norm of the linear map $\left.D f\right|_{x}$. Let $[a, b]=\{t a+(1-t) b: 0 \leqslant t \leqslant 1\}\left(a, b, \in \mathbb{R}^{n}\right)$ be a straight-line segment contained in $U$. Prove that $|f(b)-f(a)| \leqslant M\|b-a\|$, where $\|\cdot\|$ denotes the Euclidean norm on $\mathbb{R}^{n}$.

  Prove that if $U$ is an open ball and $\left.D f\right|_{x}=0$ for each $x \in U$, then $f$ is constant on $U$.

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• Paper 4, Section I, F

  #Analysis II

  2014

  Define a contraction mapping and state the contraction mapping theorem.

  Let $C[0,1]$ be the space of continuous real-valued functions on $[0,1]$ endowed with the uniform norm. Show that the map $A: C[0,1] \rightarrow C[0,1]$ defined by

  $$A f(x)=\int_{0}^{x} f(t) d t$$

  is not a contraction mapping, but that $A \circ A$ is.

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• Paper 2, Section I, F

  #Analysis II

  2014

  Define what is meant by a uniformly continuous function on a set $E \subset \mathbb{R}$.

  If $f$ and $g$ are uniformly continuous functions on $\mathbb{R}$, is the (pointwise) product $f g$ necessarily uniformly continuous on $\mathbb{R}$ ?

  Is a uniformly continuous function on $(0,1)$ necessarily bounded?

  Is $\cos (1 / x)$ uniformly continuous on $(0,1) ?$

  Justify your answers.

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• Paper 1, Section II, F

  #Analysis II

  2014

  Define what it means for two norms on a real vector space $V$ to be Lipschitz equivalent. Show that if two norms on $V$ are Lipschitz equivalent and $F \subset V$, then $F$ is closed in one norm if and only if $F$ is closed in the other norm.

  Show that if $V$ is finite-dimensional, then any two norms on $V$ are Lipschitz equivalent.

  Show that $\|f\|_{1}=\int_{0}^{1}|f(x)| d x$ is a norm on the space $C[0,1]$ of continuous realvalued functions on $[0,1]$. Is the set $S=\{f \in C[0,1]: f(1 / 2)=0\}$ closed in the norm $\|\cdot\| 1$ ?

  Determine whether or not the norm $\|\cdot\|_{1}$ is Lipschitz equivalent to the uniform $\operatorname{norm}\|\cdot\|_{\infty}$ on $C[0,1]$.

  [You may assume the Bolzano-Weierstrass theorem for sequences in $\mathbb{R}^{n}$.]

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• Paper 4, Section II, F

  #Analysis II

  2014

  Let $U \subset \mathbb{R}^{2}$ be an open set. Define what it means for a function $f: U \rightarrow \mathbb{R}$ to be differentiable at a point $\left(x_{0}, y_{0}\right) \in U$.

  Prove that if the partial derivatives $D_{1} f$ and $D_{2} f$ exist on $U$ and are continuous at $\left(x_{0}, y_{0}\right)$, then $f$ is differentiable at $\left(x_{0}, y_{0}\right)$.

  If $f$ is differentiable on $U$ must $D_{1} f, D_{2} f$ be continuous at $\left(x_{0}, y_{0}\right) ?$ Give a proof or counterexample as appropriate.

  The function $h: \mathbb{R}^{2} \rightarrow \mathbb{R}$ is defined by

  $$h(x, y)=x y \sin (1 / x) \quad \text { for } x \neq 0, \quad h(0, y)=0$$

  Determine all the points $(x, y)$ at which $h$ is differentiable.

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• Paper 3, Section II, F

  #Analysis II

  2014

  Let $f_{n}, n=1,2, \ldots$, be continuous functions on an open interval $(a, b)$. Prove that if the sequence $\left(f_{n}\right)$ converges to $f$ uniformly on $(a, b)$ then the function $f$ is continuous on $(a, b)$.

  If instead $\left(f_{n}\right)$ is only known to converge pointwise to $f$ and $f$ is continuous, must $\left(f_{n}\right)$ be uniformly convergent? Justify your answer.

  Suppose that a function $f$ has a continuous derivative on $(a, b)$ and let

  $$g_{n}(x)=n\left(f\left(x+\frac{1}{n}\right)-f(x)\right)$$

  Stating clearly any standard results that you require, show that the functions $g_{n}$ converge uniformly to $f^{\prime}$ on each interval $[\alpha, \beta] \subset(a, b)$.

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• Paper 2, Section II, $12 \mathrm{~F}$

  #Analysis II

  2014

  Let $X, Y$ be subsets of $\mathbb{R}^{n}$ and define $X+Y=\{x+y: x \in X, y \in Y\}$. For each of the following statements give a proof or a counterexample (with justification) as appropriate.

  (i) If each of $X, Y$ is bounded and closed, then $X+Y$ is bounded and closed.

  (ii) If $X$ is bounded and closed and $Y$ is closed, then $X+Y$ is closed.

  (iii) If $X, Y$ are both closed, then $X+Y$ is closed.

  (iv) If $X$ is open and $Y$ is closed, then $X+Y$ is open.

  [The Bolzano-Weierstrass theorem in $\mathbb{R}^{n}$ may be assumed without proof.]

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• Paper 4, Section I, G

  #Complex Analysis

  2014

  Let $f$ be an entire function. State Cauchy's Integral Formula, relating the $n$th derivative of $f$ at a point $z$ with the values of $f$ on a circle around $z$.

  State Liouville's Theorem, and deduce it from Cauchy's Integral Formula.

  Let $f$ be an entire function, and suppose that for some $k$ we have that $|f(z)| \leqslant|z|^{k}$ for all $z$. Prove that $f$ is a polynomial.

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• Paper 1, Section II, B

  #Complex Analysis or Complex Methods

  2014

  By choice of a suitable contour show that for $a>b>0$

  $$\int_{0}^{2 \pi} \frac{\sin ^{2} \theta d \theta}{a+b \cos \theta}=\frac{2 \pi}{b^{2}}\left[a-\sqrt{a^{2}-b^{2}}\right]$$

  Hence evaluate

  $$\int_{0}^{1} \frac{\left(1-x^{2}\right)^{1 / 2} x^{2} d x}{1+x^{2}}$$

  using the substitution $x=\cos (\theta / 2)$.

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• Paper 2, Section II, B

  #Complex Analysis or Complex Methods

  2014

  By considering a rectangular contour, show that for $0<a<1$ we have

  $$\int_{-\infty}^{\infty} \frac{e^{a x}}{e^{x}+1} d x=\frac{\pi}{\sin \pi a}$$

  Hence evaluate

  $$\int_{0}^{\infty} \frac{d t}{t^{5 / 6}(1+t)}$$

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• Paper 3, Section II, G

  #Complex Analysis

  2014

  State the Residue Theorem precisely.

  Let $D$ be a star-domain, and let $\gamma$ be a closed path in $D$. Suppose that $f$ is a holomorphic function on $D$, having no zeros on $\gamma$. Let $N$ be the number of zeros of $f$ inside $\gamma$, counted with multiplicity (i.e. order of zero and winding number). Show that

  $$N=\frac{1}{2 \pi i} \int_{\gamma} \frac{f^{\prime}(z)}{f(z)} d z$$

  [The Residue Theorem may be used without proof.]

  Now suppose that $g$ is another holomorphic function on $D$, also having no zeros on $\gamma$ and with $|g(z)|<|f(z)|$ on $\gamma$. Explain why, for any $0 \leqslant t \leqslant 1$, the expression

  $$I(t)=\int_{\gamma} \frac{f^{\prime}(z)+\operatorname{tg}^{\prime}(z)}{f(z)+\operatorname{tg}(z)} d z$$

  is well-defined. By considering the behaviour of the function $I(t)$ as $t$ varies, deduce Rouché's Theorem.

  For each $n$, let $p_{n}$ be the polynomial $\sum_{k=0}^{n} \frac{z^{k}}{k !}$. Show that, as $n$ tends to infinity, the smallest modulus of the roots of $p_{n}$ also tends to infinity.

  [You may assume any results on convergence of power series, provided that they are stated clearly.]

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• Paper 1, Section I, B

  #Complex Analysis or Complex Methods

  2014

  Let $f(z)$ be an analytic/holomorphic function defined on an open set $D$, and let $z_{0} \in D$ be a point such that $f^{\prime}\left(z_{0}\right) \neq 0$. Show that the transformation $w=f(z)$ preserves the angle between smooth curves intersecting at $z_{0}$. Find such a transformation $w=f(z)$ that maps the second quadrant of the unit disc (i.e. $|z|<1, \pi / 2<\arg (z)<\pi)$ to the region in the first quadrant of the complex plane where $|w|>1$ (i.e. the region in the first quadrant outside the unit circle).

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• Paper 3, Section I, B

  #Complex Methods

  2014

  Find the most general cubic form

  $$u(x, y)=a x^{3}+b x^{2} y+c x y^{2}+d y^{3}$$

  which satisfies Laplace's equation, where $a, b, c$ and $d$ are all real. Hence find an analytic function $f(z)=f(x+i y)$ which has such a $u$ as its real part.

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• Paper 4, Section II, B

  #Complex Methods

  2014

  Find the Laplace transforms of $t^{n}$ for $n$ a positive integer and $H(t-a)$ where $a>0$ and $H(t)$ is the Heaviside step function.

  Consider a semi-infinite string which is initially at rest and is fixed at one end. The string can support wave-like motions, and for $t>0$ it is allowed to fall under gravity. Therefore the deflection $y(x, t)$ from its initial location satisfies

  $$\frac{\partial^{2}}{\partial t^{2}} y=c^{2} \frac{\partial^{2}}{\partial x^{2}} y+g \quad \text { for } \quad x>0, t>0$$

  with

  $$y(0, t)=y(x, 0)=\frac{\partial}{\partial t} y(x, 0)=0 \quad \text { and } \quad y(x, t) \rightarrow \frac{g t^{2}}{2} \text { as } x \rightarrow \infty$$

  where $g$ is a constant. Use Laplace transforms to find $y(x, t)$.

  [The convolution theorem for Laplace transforms may be quoted without proof.]

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• Paper 2, Section I, A

  #Electromagnetism

  2014

  Starting from Maxwell's equations, deduce that

  $$\frac{d \Phi}{d t}=-\mathcal{E}$$

  for a moving circuit $C$, where $\Phi$ is the flux of $\mathbf{B}$ through the circuit and where the electromotive force $\mathcal{E}$ is defined to be

  $$\mathcal{E}=\oint_{\mathcal{C}}(\mathbf{E}+\mathbf{v} \times \mathbf{B}) \cdot \mathbf{d} \mathbf{r}$$

  where $\mathbf{v}=\mathbf{v}(\mathbf{r})$ denotes the velocity of a point $\mathbf{r}$ on $C$.

  [Hint: Consider the closed surface consisting of the surface $S(t)$ bounded by $C(t)$, the surface $S(t+\delta t)$ bounded by $C(t+\delta t)$ and the surface $S^{\prime}$ stretching from $C(t)$ to $C(t+\delta t)$. Show that the flux of $\mathbf{B}$ through $S^{\prime}$ is $-\delta t \oint_{C} \mathbf{B} \cdot(\mathbf{v} \times \mathbf{d r})$.]

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• Paper 4, Section I, A

  #Electromagnetism

  2014

  A continuous wire of resistance $R$ is wound around a very long right circular cylinder of radius $a$, and length $l$ (long enough so that end effects can be ignored). There are $N \gg 1$ turns of wire per unit length, wound in a spiral of very small pitch. Initially, the magnetic field $\mathbf{B}$ is $\mathbf{0}$.

  Both ends of the coil are attached to a battery of electromotance $\mathcal{E}_{0}$ at $t=0$, which induces a current $I(t)$. Use Ampère's law to derive $\mathbf{B}$ inside and outside the cylinder when the displacement current may be neglected. Write the self-inductance of the coil $L$ in terms of the quantities given above. Using Ohm's law and Faraday's law of induction, find $I(t)$ explicitly in terms of $\mathcal{E}_{0}, R, L$ and $t$.

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• Paper 1, Section II, A

  #Electromagnetism

  2014

  The region $z<0$ is occupied by an ideal earthed conductor and a point charge $q$ with mass $m$ is held above it at $(0,0, d)$.

  (i) What are the boundary conditions satisfied by the electric field $\mathbf{E}$ on the surface of the conductor?

  (ii) Consider now a system without the conductor mentioned above. A point charge $q$ with mass $m$ is held at $(0,0, d)$, and one of charge $-q$ is held at $(0,0,-d)$. Show that the boundary condition on $\mathbf{E}$ at $z=0$ is identical to the answer to (i). Explain why this represents the electric field due to the charge at $(0,0, d)$ under the influence of the conducting boundary.

  (iii) The original point charge in (i) is released with zero initial velocity. Find the time taken for the point charge to reach the plane (ignoring gravity).

  [You may assume that the force on the point charge is equal to $m d^{2} \mathbf{x} / d t^{2}$, where $\mathbf{x}$ is the position vector of the charge, and $t$ is time.]

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• Paper 3, Section II, A

  #Electromagnetism

  2014

  (i) Consider charges $-q$ at $\pm \mathbf{d}$ and $2 q$ at $(0,0,0)$. Write down the electric potential.

  (ii) Take $\mathbf{d}=(0,0, d)$. A quadrupole is defined in the limit that $q \rightarrow \infty, d \rightarrow 0$ such that $q d^{2}$ tends to a constant $p$. Find the quadrupole's potential, showing that it is of the form

  $$\phi(\mathbf{r})=A \frac{\left(r^{2}+C z^{D}\right)}{r^{B}}$$

  where $r=|\mathbf{r}|$. Determine the constants $A, B, C$ and $D$.

  (iii) The quadrupole is fixed at the origin. At time $t=0$ a particle of charge $-Q(Q$ has the same sign as $q)$ and mass $m$ is at $(1,0,0)$ travelling with velocity $d \mathbf{r} / d t=(-\kappa, 0,0)$, where

  $$\kappa=\sqrt{\frac{Q p}{2 \pi \epsilon_{0} m}} .$$

  Neglecting gravity, find the time taken for the particle to reach the quadrupole in terms of $\kappa$, given that the force on the particle is equal to $m d^{2} \mathbf{r} / d t^{2}$.

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• Paper 2, Section II, A

  #Electromagnetism

  2014

  What is the relationship between the electric field $\mathbf{E}$ and the charge per unit area $\sigma$ on the surface of a perfect conductor?

  Consider a charge distribution $\rho(\mathbf{r})$ distributed with potential $\phi(\mathbf{r})$ over a finite volume $V$ within which there is a set of perfect conductors with charges $Q_{i}$, each at a potential $\phi_{i}$ (normalised such that the potential at infinity is zero). Using Maxwell's equations and the divergence theorem, derive a relationship between the electrostatic energy $W$ and a volume integral of an explicit function of the electric field $\mathbf{E}$, where

  $$W=\frac{1}{2} \int_{V} \rho \phi d \tau+\frac{1}{2} \sum_{i} Q_{i} \phi_{i}$$

  Consider $N$ concentric perfectly conducting spherical shells. Shell $n$ has radius $r_{n}$ (where $r_{n}>r_{n-1}$ ) and charge $q$ for $n=1$, and charge $2(-1)^{(n+1)} q$ for $n>1$. Show that

  $$W \propto \frac{1}{r_{1}},$$

  and determine the constant of proportionality.

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• Paper 1, Section I, B

  #Fluid Dynamics

  2014

  Constant density viscous fluid with dynamic viscosity $\mu$ flows in a two-dimensional horizontal channel of depth $h$. There is a constant pressure gradient $G>0$ in the horizontal $x$-direction. The upper horizontal boundary at $y=h$ is driven at constant horizontal speed $U>0$, with the lower boundary being held at rest. Show that the steady fluid velocity $u$ in the $x$-direction is

  $$u=\frac{-G}{2 \mu} y(h-y)+\frac{U y}{h}$$

  Show that it is possible to have $d u / d y<0$ at some point in the flow for sufficiently large pressure gradient. Derive a relationship between $G$ and $U$ so that there is no net volume flux along the channel. For the flow with no net volume flux, sketch the velocity profile.

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• Paper 2, Section I, B

  #Fluid Dynamics

  2014

  Consider the steady two-dimensional fluid velocity field

  $$\mathbf{u}=\left(\begin{array}{l} u \\ v \end{array}\right)=\left(\begin{array}{ll} \epsilon & -\gamma \\ \gamma & -\epsilon \end{array}\right)\left(\begin{array}{l} x \\ y \end{array}\right)$$

  where $\epsilon \geqslant 0$ and $\gamma \geqslant 0$. Show that the fluid is incompressible. The streamfunction $\psi$ is defined by $\mathbf{u}=\boldsymbol{\nabla} \times \boldsymbol{\Psi}$, where $\boldsymbol{\Psi}=(0,0, \psi)$. Show that $\psi$ is given by

  $$\psi=\epsilon x y-\frac{\gamma}{2}\left(x^{2}+y^{2}\right)$$

  Hence show that the streamlines are defined by

  $$(\epsilon-\gamma)(x+y)^{2}-(\epsilon+\gamma)(x-y)^{2}=C$$

  for $C$ a constant. For each of the three cases below, sketch the streamlines and briefly describe the flow. (i) $\epsilon=1, \gamma=0$, (ii) $\epsilon=0, \gamma=1$, (iii) $\epsilon=1, \gamma=1$.

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• Paper 1, Section II, B

  #Fluid Dynamics

  2014

  Consider the purely two-dimensional steady flow of an inviscid incompressible constant density fluid in the absence of body forces. For velocity $\mathbf{u}$, the vorticity is $\boldsymbol{\nabla} \times \mathbf{u}=\boldsymbol{\omega}=(0,0, \omega)$. Show that

  $$\mathbf{u} \times \boldsymbol{\omega}=\boldsymbol{\nabla}\left[\frac{p}{\rho}+\frac{1}{2}|\mathbf{u}|^{2}\right]$$

  where $p$ is the pressure and $\rho$ is the fluid density. Hence show that, if $\omega$ is a constant in both space and time,

  $$\frac{1}{2}|\mathbf{u}|^{2}+\omega \psi+\frac{p}{\rho}=C,$$

  where $C$ is a constant and $\psi$ is the streamfunction. Here, $\psi$ is defined by $\mathbf{u}=\boldsymbol{\nabla} \times \boldsymbol{\Psi}$, where $\boldsymbol{\Psi}=(0,0, \psi)$.

  Fluid in the annular region $a<r<2 a$ has constant (in both space and time) vorticity $\omega$. The streamlines are concentric circles, with the fluid speed zero on $r=2 a$ and $V>0$ on $r=a$. Calculate the velocity field, and hence show that

  $$\omega=\frac{-2 V}{3 a}$$

  Deduce that the pressure difference between the outer and inner edges of the annular region is

  $$\Delta p=\left(\frac{15-16 \ln 2}{18}\right) \rho V^{2}$$

  [Hint: Note that in cylindrical polar coordinates $(r, \phi, z)$, the curl of a vector field $\mathbf{A}(r, \phi)=[a(r, \phi), b(r, \phi), c(r, \phi)]$ is

  $$\boldsymbol{\nabla} \times \mathbf{A}=\left[\frac{1}{r} \frac{\partial c}{\partial \phi},-\frac{\partial c}{\partial r}, \frac{1}{r}\left(\frac{\partial(r b)}{\partial r}-\frac{\partial a}{\partial \phi}\right)\right]$$

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• Paper 4, Section II, B

  #Fluid Dynamics

  2014

  Consider a layer of fluid of constant density $\rho$ and equilibrium depth $h_{0}$ in a rotating frame of reference, rotating at constant angular velocity $\Omega$ about the vertical $z$-axis. The equations of motion are

  $$\begin{aligned} \frac{\partial u}{\partial t}-f v &=-\frac{1}{\rho} \frac{\partial p}{\partial x} \\ \frac{\partial v}{\partial t}+f u &=-\frac{1}{\rho} \frac{\partial p}{\partial y} \\ 0 &=-\frac{\partial p}{\partial z}-\rho g \end{aligned}$$

  where $p$ is the fluid pressure, $u$ and $v$ are the fluid velocities in the $x$-direction and $y$ direction respectively, $f=2 \Omega$, and $g$ is the constant acceleration due to gravity. You may also assume that the horizontal extent of the layer is sufficiently large so that the layer may be considered to be shallow, such that vertical velocities may be neglected.

  By considering mass conservation, show that the depth $h(x, y, t)$ of the layer satisfies

  $$\frac{\partial h}{\partial t}+\frac{\partial}{\partial x}(h u)+\frac{\partial}{\partial y}(h v)=0 .$$

  Now assume that $h=h_{0}+\eta(x, y, t)$, where $|\eta| \ll h_{0}$. Show that the (linearised) potential vorticity $\mathbf{Q}=Q \hat{\mathbf{z}}$, defined by

  $$Q=\zeta-\eta \frac{f}{h_{0}}, \text { where } \zeta=\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}$$

  and $\hat{\mathbf{z}}$ is the unit vector in the vertical $z$-direction, is a constant in time, i.e. $Q=Q_{0}(x, y)$.

  When $Q_{0}=0$ everywhere, establish that the surface perturbation $\eta$ satisfies

  $$\frac{\partial^{2} \eta}{\partial t^{2}}-g h_{0}\left(\frac{\partial^{2} \eta}{\partial x^{2}}+\frac{\partial^{2} \eta}{\partial y^{2}}\right)+f^{2} \eta=0$$

  and show that this equation has wave-like solutions $\eta=\eta_{0} \cos [k(x-c t)]$ when $c$ and $k$ are related through a dispersion relation to be determined. Show that, to leading order, the trajectories of fluid particles for these waves are ellipses. Assuming that $\eta_{0}>0, k>0$, $c>0$ and $f>0$, sketch the fluid velocity when $k(x-c t)=n \pi / 2$ for $n=0,1,2,3$.

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• Paper 3, Section II, B

  #Fluid Dynamics

  2014

  A bubble of gas occupies the spherical region $r \leqslant R(t)$, and an incompressible irrotational liquid of constant density $\rho$ occupies the outer region $r \geqslant R$, such that as $r \rightarrow \infty$ the liquid is at rest with constant pressure $p_{\infty}$. Briefly explain why it is appropriate to use a velocity potential $\phi(r, t)$ to describe the liquid velocity u.

  By applying continuity of velocity across the gas-liquid interface, show that the liquid pressure (for $r \geqslant R$ ) satisfies

  $$\frac{p}{\rho}+\frac{1}{2}\left(\frac{R^{2} \dot{R}}{r^{2}}\right)^{2}-\frac{1}{r} \frac{d}{d t}\left(R^{2} \dot{R}\right)=\frac{p_{\infty}}{\rho}, \quad \text { where } \dot{R}=\frac{d R}{d t} .$$

  Show that the excess pressure $p_{s}-p_{\infty}$ at the bubble surface $r=R$ is

  $$p_{s}-p_{\infty}=\frac{\rho}{2}\left(3 \dot{R}^{2}+2 R \ddot{R}\right), \quad \text { where } \ddot{R}=\frac{d^{2} R}{d t^{2}}$$

  and hence that

  $$p_{s}-p_{\infty}=\frac{\rho}{2 R^{2}} \frac{d}{d R}\left(R^{3} \dot{R}^{2}\right)$$

  The pressure $p_{g}(t)$ inside the gas bubble satisfies the equation of state

  $$p_{g} V^{4 / 3}=C$$

  where $C$ is a constant, and $V(t)$ is the bubble volume. At time $t=0$ the bubble is at rest with radius $R=a$. If the bubble then expands and comes to rest at $R=2 a$, determine the required gas pressure $p_{0}$ at $t=0$ in terms of $p_{\infty}$.

  [You may assume that there is contact between liquid and gas for all time, that all motion is spherically symmetric about the origin $r=0$, and that there is no body force. You may also assume Bernoulli's integral of the equation of motion to determine the liquid pressure

  $$\frac{p}{\rho}+\frac{\partial \phi}{\partial t}+\frac{1}{2}|\nabla \phi|^{2}=A(t)$$

  where $\phi(r, t)$ is the velocity potential.]

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• Paper 1, Section I, F

  #Geometry

  2014

  Determine the second fundamental form of a surface in $\mathbb{R}^{3}$ defined by the parametrisation

  $$\sigma(u, v)=((a+b \cos u) \cos v,(a+b \cos u) \sin v, b \sin u)$$

  for $0<u<2 \pi, 0<v<2 \pi$, with some fixed $a>b>0$. Show that the Gaussian curvature $K(u, v)$ of this surface takes both positive and negative values.

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• Paper 3, Section I, F

  #Geometry

  2014

  Let $f(x)=A x+b$ be an isometry $\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$, where $A$ is an $n \times n$ matrix and $b \in \mathbb{R}^{n}$. What are the possible values of $\operatorname{det} A$ ?

  Let $I$ denote the $n \times n$ identity matrix. Show that if $n=2$ and $\operatorname{det} A>0$, but $A \neq I$, then $f$ has a fixed point. Must $f$ have a fixed point if $n=3$ and $\operatorname{det} A>0$, but $A \neq I ?$ Justify your answer.

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• Paper 3, Section II, F

  #Geometry

  2014

  Let $\mathcal{T}$ be a decomposition of the two-dimensional sphere into polygonal domains, with every polygon having at least three edges. Let $V, E$, and $F$ denote the numbers of vertices, edges and faces of $\mathcal{T}$, respectively. State Euler's formula. Prove that $2 E \geqslant 3 F$.

  Suppose that at least three edges meet at every vertex of $\mathcal{T}$. Let $F_{n}$ be the number of faces of $\mathcal{T}$ that have exactly $n$ edges $(n \geqslant 3)$ and let $V_{m}$ be the number of vertices at which exactly $m$ edges meet $(m \geqslant 3)$. Is it possible for $\mathcal{T}$ to have $V_{3}=F_{3}=0$ ? Justify your answer.

  By expressing $6 F-\sum_{n} n F_{n}$ in terms of the $V_{j}$, or otherwise, show that $\mathcal{T}$ has at least four faces that are triangles, quadrilaterals and/or pentagons.

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• Paper 2, Section II, F

  #Geometry

  2014

  Let $H=\{x+i y: x, y \in \mathbb{R}, y>0\} \subset \mathbb{C}$ be the upper half-plane with a hyperbolic metric $g=\frac{d x^{2}+d y^{2}}{y^{2}}$. Prove that every hyperbolic circle $C$ in $H$ is also a Euclidean circle. Is the centre of $C$ as a hyperbolic circle always the same point as the centre of $C$ as a Euclidean circle? Give a proof or counterexample as appropriate.

  Let $A B C$ and $A^{\prime} B^{\prime} C^{\prime}$ be two hyperbolic triangles and denote the hyperbolic lengths of their sides by $a, b, c$ and $a^{\prime}, b^{\prime}, c^{\prime}$, respectively. Show that if $a=a^{\prime}, b=b^{\prime}$ and $c=c^{\prime}$, then there is a hyperbolic isometry taking $A B C$ to $A^{\prime} B^{\prime} C^{\prime}$. Is there always such an isometry if instead the triangles have one angle the same and $a=a^{\prime}, b=b^{\prime} ?$ Justify your answer.

  [Standard results on hyperbolic isometries may be assumed, provided they are clearly stated.]

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• Paper 4, Section II, F

  #Geometry

  2014

  Define an embedded parametrised surface in $\mathbb{R}^{3}$. What is the Riemannian metric induced by a parametrisation? State, in terms of the Riemannian metric, the equations defining a geodesic curve $\gamma:(0,1) \rightarrow S$, assuming that $\gamma$ is parametrised by arc-length.

  Let $S$ be a conical surface

  $$S=\left\{(x, y, z) \in \mathbb{R}^{3}: 3\left(x^{2}+y^{2}\right)=z^{2}, \quad z>0\right\}$$

  Using an appropriate smooth parametrisation, or otherwise, prove that $S$ is locally isometric to the Euclidean plane. Show that any two points on $S$ can be joined by a geodesic. Is this geodesic always unique (up to a reparametrisation)? Justify your answer.

  [The expression for the Euclidean metric in polar coordinates on $\mathbb{R}^{2}$ may be used without proof.]

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• Paper 3, Section I, E

  #Groups, Rings and Modules

  2014

  State and prove Hilbert's Basis Theorem.

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• Paper 4, Section I, E

  #Groups, Rings and Modules

  2014

  Let $G$ be the abelian group generated by elements $a, b$ and $c$ subject to the relations: $3 a+6 b+3 c=0,9 b+9 c=0$ and $-3 a+3 b+6 c=0$. Express $G$ as a product of cyclic groups. Hence determine the number of elements of $G$ of order 3 .

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• Paper 2, Section I, $2 E$

  #Groups, Rings and Modules

  2014

  List the conjugacy classes of $A_{6}$ and determine their sizes. Hence prove that $A_{6}$ is simple.

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• Paper 1, Section II, E

  #Groups, Rings and Modules

  2014

  Let $G$ be a finite group and $p$ a prime divisor of the order of $G$. Give the definition of a Sylow $p$-subgroup of $G$, and state Sylow's theorems.

  Let $p$ and $q$ be distinct primes. Prove that a group of order $p^{2} q$ is not simple.

  Let $G$ be a finite group, $H$ a normal subgroup of $G$ and $P$ a Sylow $p$-subgroup of H. Let $N_{G}(P)$ denote the normaliser of $P$ in $G$. Prove that if $g \in G$ then there exist $k \in N_{G}(P)$ and $h \in H$ such that $g=k h$.

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• Paper 4, Section II, E

  #Groups, Rings and Modules

  2014

  (a) Consider the four following types of rings: Principal Ideal Domains, Integral Domains, Fields, and Unique Factorisation Domains. Arrange them in the form $A \Longrightarrow$ $B \Longrightarrow C \Longrightarrow D$ (where $A \Longrightarrow B$ means if a ring is of type $A$ then it is of type $B$ )

  Prove that these implications hold. [You may assume that irreducibles in a Principal Ideal Domain are prime.] Provide examples, with brief justification, to show that these implications cannot be reversed.

  (b) Let $R$ be a ring with ideals $I$ and $J$ satisfying $I \subseteq J$. Define $K$ to be the set $\{r \in R: r J \subseteq I\}$. Prove that $K$ is an ideal of $R$. If $J$ and $K$ are principal, prove that $I$ is principal.

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• Paper 3, Section II, E

  #Groups, Rings and Modules

  2014

  Let $R$ be a ring, $M$ an $R$-module and $S=\left\{m_{1}, \ldots, m_{k}\right\}$ a subset of $M$. Define what it means to say $S$ spans $M$. Define what it means to say $S$ is an independent set.

  We say $S$ is a basis for $M$ if $S$ spans $M$ and $S$ is an independent set. Prove that the following two statements are equivalent.

  1. $S$ is a basis for $M$.

  2. Every element of $M$ is uniquely expressible in the form $r_{1} m_{1}+\cdots+r_{k} m_{k}$ for some $r_{1}, \ldots, r_{k} \in R$.

  We say $S$ generates $M$ freely if $S$ spans $M$ and any map $\Phi: S \rightarrow N$, where $N$ is an $R$-module, can be extended to an $R$-module homomorphism $\Theta: M \rightarrow N$. Prove that $S$ generates $M$ freely if and only if $S$ is a basis for $M$.

  Let $M$ be an $R$-module. Are the following statements true or false? Give reasons.

  (i) If $S$ spans $M$ then $S$ necessarily contains an independent spanning set for $M$.

  (ii) If $S$ is an independent subset of $M$ then $S$ can always be extended to a basis for $M$.

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• Paper 2, Section II, 11E

  #Groups, Rings and Modules

  2014

  Prove that every finite integral domain is a field.

  Let $F$ be a field and $f$ an irreducible polynomial in the polynomial ring $F[X]$. Prove that $F[X] /(f)$ is a field, where $(f)$ denotes the ideal generated by $f$.

  Hence construct a field of 4 elements, and write down its multiplication table.

  Construct a field of order 9 .

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• Paper 4, Section I, G

  #Linear Algebra

  2014

  Let $V$ denote the vector space of all real polynomials of degree at most 2 . Show that

  $$(f, g)=\int_{-1}^{1} f(x) g(x) d x$$

  defines an inner product on $V$.

  Find an orthonormal basis for $V$.

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• Paper 2, Section I, G

  #Linear Algebra

  2014

  State and prove the Rank-Nullity Theorem.

  Let $\alpha$ be a linear map from $\mathbb{R}^{5}$ to $\mathbb{R}^{3}$. What are the possible dimensions of the kernel of $\alpha$ ? Justify your answer.

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• Paper 1, Section I, G

  #Linear Algebra

  2014

  State and prove the Steinitz Exchange Lemma. Use it to prove that, in a finitedimensional vector space: any two bases have the same size, and every linearly independent set extends to a basis.

  Let $e_{1}, \ldots, e_{n}$ be the standard basis for $\mathbb{R}^{n}$. Is $e_{1}+e_{2}, e_{2}+e_{3}, e_{3}+e_{1}$ a basis for $\mathbb{R}^{3} ?$ Is $e_{1}+e_{2}, e_{2}+e_{3}, e_{3}+e_{4}, e_{4}+e_{1}$ a basis for $\mathbb{R}^{4} ?$ Justify your answers.

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• Paper 1, Section II, G

  #Linear Algebra

  2014

  Let $V$ be an $n$-dimensional real vector space, and let $T$ be an endomorphism of $V$. We say that $T$ acts on a subspace $W$ if $T(W) \subset W$.

  (i) For any $x \in V$, show that $T$ acts on the linear span of $\left\{x, T(x), T^{2}(x), \ldots, T^{n-1}(x)\right\}$.

  (ii) If $\left\{x, T(x), T^{2}(x), \ldots, T^{n-1}(x)\right\}$ spans $V$, show directly (i.e. without using the CayleyHamilton Theorem) that $T$ satisfies its own characteristic equation.

  (iii) Suppose that $T$ acts on a subspace $W$ with $W \neq\{0\}$ and $W \neq V$. Let $e_{1}, \ldots, e_{k}$ be a basis for $W$, and extend to a basis $e_{1}, \ldots, e_{n}$ for $V$. Describe the matrix of $T$ with respect to this basis.

  (iv) Using (i), (ii) and (iii) and induction, give a proof of the Cayley-Hamilton Theorem.

  [Simple properties of determinants may be assumed without proof.]

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• Paper 4, Section II, G

  #Linear Algebra

  2014

  Let $V$ be a real vector space. What is the dual $V^{*}$ of $V ?$ If $e_{1}, \ldots, e_{n}$ is a basis for $V$, define the dual basis $e_{1}^{*}, \ldots, e_{n}^{*}$ for $V^{*}$, and show that it is indeed a basis for $V^{*}$.

  [No result about dimensions of dual spaces may be assumed.]

  For a subspace $U$ of $V$, what is the annihilator of $U$ ? If $V$ is $n$-dimensional, how does the dimension of the annihilator of $U$ relate to the dimension of $U$ ?

  Let $\alpha: V \rightarrow W$ be a linear map between finite-dimensional real vector spaces. What is the dual map $\alpha^{*}$ ? Explain why the rank of $\alpha^{*}$ is equal to the rank of $\alpha$. Prove that the kernel of $\alpha^{*}$ is the annihilator of the image of $\alpha$, and also that the image of $\alpha^{*}$ is the annihilator of the kernel of $\alpha$.

  [Results about the matrices representing a map and its dual may be used without proof, provided they are stated clearly.]

  Now let $V$ be the vector space of all real polynomials, and define elements $L_{0}, L_{1}, \ldots$ of $V^{*}$ by setting $L_{i}(p)$ to be the coefficient of $X^{i}$ in $p$ (for each $p \in V$ ). Do the $L_{i}$ form a basis for $V^{*}$ ?

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• Paper 3, Section II, G

  #Linear Algebra

  2014

  Let $q$ be a nonsingular quadratic form on a finite-dimensional real vector space $V$. Prove that we may write $V=P \bigoplus N$, where the restriction of $q$ to $P$ is positive definite, the restriction of $q$ to $N$ is negative definite, and $q(x+y)=q(x)+q(y)$ for all $x \in P$ and $y \in N$. [No result on diagonalisability may be assumed.]

  Show that the dimensions of $P$ and $N$ are independent of the choice of $P$ and $N$. Give an example to show that $P$ and $N$ are not themselves uniquely defined.

  Find such a decomposition $V=P \bigoplus N$ when $V=\mathbb{R}^{3}$ and $q$ is the quadratic form $q((x, y, z))=x^{2}+2 y^{2}-2 x y-2 x z$

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• Paper 2, Section II, G

  #Linear Algebra

  2014

  Define the determinant of an $n \times n$ complex matrix $A$. Explain, with justification, how the determinant of $A$ changes when we perform row and column operations on $A$.

  Let $A, B, C$ be complex $n \times n$ matrices. Prove the following statements. (i) $\operatorname{det}\left(\begin{array}{cc}A & C \\ 0 & B\end{array}\right)=\operatorname{det} A \operatorname{det} B$. (ii) $\operatorname{det}\left(\begin{array}{cc}A & -B \\ B & A\end{array}\right)=\operatorname{det}(A+i B) \operatorname{det}(A-i B)$.

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• Paper 4, Section I, H

  #Markov Chains

  2014

  Let $\left(X_{n}: n \geqslant 0\right)$ be a homogeneous Markov chain with state space $S$ and transition $\operatorname{matrix} P=\left(p_{i, j}: i, j \in S\right)$.

  (a) Let $W_{n}=X_{2 n}, n=0,1,2, \ldots$ Show that $\left(W_{n}: n \geqslant 0\right)$ is a Markov chain and give its transition matrix. If $\lambda_{i}=\mathbb{P}\left(X_{0}=i\right), i \in S$, find $\mathbb{P}\left(W_{1}=0\right)$ in terms of the $\lambda_{i}$ and the $p_{i, j}$.

  [Results from the course may be quoted without proof, provided they are clearly stated.]

  (b) Suppose that $S=\{-1,0,1\}, p_{0,1}=p_{-1,-1}=0$ and $p_{-1,0} \neq p_{1,0}$. Let $Y_{n}=\left|X_{n}\right|$, $n=0,1,2, \ldots$ In terms of the $p_{i, j}$, find

  (i) $\mathbb{P}\left(Y_{n+1}=0 \mid Y_{n}=1, Y_{n-1}=0\right)$ and

  (ii) $\mathbb{P}\left(Y_{n+1}=0 \mid Y_{n}=1, Y_{n-1}=1, Y_{n-2}=0\right)$.

  What can you conclude about whether or not $\left(Y_{n}: n \geqslant 0\right)$ is a Markov chain?

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• Paper 3, Section I, H

  #Markov Chains

  2014

  Let $\left(X_{n}: n \geqslant 0\right)$ be a homogeneous Markov chain with state space $S$. For $i, j$ in $S$ let $p_{i, j}(n)$ denote the $n$-step transition probability $\mathbb{P}\left(X_{n}=j \mid X_{0}=i\right)$.

  (i) Express the $(m+n)$-step transition probability $p_{i, j}(m+n)$ in terms of the $n$-step and $m$-step transition probabilities.

  (ii) Write $i \rightarrow j$ if there exists $n \geqslant 0$ such that $p_{i, j}(n)>0$, and $i \leftrightarrow j$ if $i \rightarrow j$ and $j \rightarrow i$. Prove that if $i \leftrightarrow j$ and $i \neq j$ then either both $i$ and $j$ are recurrent or both $i$ and $j$ are transient. [You may assume that a state $i$ is recurrent if and only if $\sum_{n=0}^{\infty} p_{i, i}(n)=\infty$, and otherwise $i$ is transient.]

  (iii) A Markov chain has state space $\{0,1,2,3\}$ and transition matrix

  $$\left(\begin{array}{cccc} \frac{1}{2} & \frac{1}{3} & 0 & \frac{1}{6} \\ 0 & \frac{3}{4} & 0 & \frac{1}{4} \\ \frac{1}{2} & \frac{1}{2} & 0 & 0 \\ \frac{1}{2} & 0 & 0 & \frac{1}{2} \end{array}\right)$$

  For each state $i$, determine whether $i$ is recurrent or transient. [Results from the course may be quoted without proof, provided they are clearly stated.]

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• Paper 1, Section II, 20H

  #Markov Chains

  2014

  Consider a homogeneous Markov chain $\left(X_{n}: n \geqslant 0\right)$ with state space $S$ and transition $\operatorname{matrix} P=\left(p_{i, j}: i, j \in S\right)$. For a state $i$, define the terms aperiodic, positive recurrent and ergodic.

  Let $S=\{0,1,2, \ldots\}$ and suppose that for $i \geqslant 1$ we have $p_{i, i-1}=1$ and

  $$p_{0,0}=0, p_{0, j}=p q^{j-1}, j=1,2, \ldots,$$

  where $p=1-q \in(0,1)$. Show that this Markov chain is irreducible.

  Let $T_{0}=\inf \left\{n \geqslant 1: X_{n}=0\right\}$ be the first passage time to 0 . Find $\mathbb{P}\left(T_{0}=n \mid X_{0}=0\right)$ and show that state 0 is ergodic.

  Find the invariant distribution $\pi$ for this Markov chain. Write down:

  (i) the mean recurrence time for state $i, i \geqslant 1$;

  (ii) $\lim _{n \rightarrow \infty} \mathbb{P}\left(X_{n} \neq 0 \mid X_{0}=0\right)$.

  [Results from the course may be quoted without proof, provided they are clearly stated.]

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• Paper 2, Section II, H

  #Markov Chains

  2014

  Let $\left(X_{n}: n \geqslant 0\right)$ be a homogeneous Markov chain with state space $\mathrm{S}$ and transition matrix $P=\left(p_{i, j}: i, j \in S\right)$. For $A \subseteq S$, let

  $$H^{A}=\inf \left\{n \geqslant 0: X_{n} \in A\right\} \text { and } h_{i}^{A}=\mathbb{P}\left(H^{A}<\infty \mid X_{0}=i\right), i \in S$$

  Prove that $h^{A}=\left(h_{i}^{A}: i \in S\right)$ is the minimal non-negative solution to the equations

  $$h_{i}^{A}= \begin{cases}1 & \text { for } i \in A \\ \sum_{j \in S} p_{i, j} h_{j}^{A} & \text { otherwise. }\end{cases}$$

  Three people $A, B$ and $C$ play a series of two-player games. In the first game, two people play and the third person sits out. Any subsequent game is played between the winner of the previous game and the person sitting out the previous game. The overall winner of the series is the first person to win two consecutive games. The players are evenly matched so that in any game each of the two players has probability $\frac{1}{2}$ of winning the game, independently of all other games. For $n=1,2, \ldots$, let $X_{n}$ be the ordered pair consisting of the winners of games $n$ and $n+1$. Thus the state space is $\{A A, A B, A C, B A, B B, B C, C A, C B, C C\}$, and, for example, $X_{1}=A C$ if $A$ wins the first game and $C$ wins the second.

  The first game is between $A$ and $B$. Treating $A A, B B$ and $C C$ as absorbing states, or otherwise, find the probability of winning the series for each of the three players.

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• Paper 4, Section I, D

  #Methods

  2014

  Consider the ordinary differential equation

  $$\frac{d^{2} \psi}{d z^{2}}-\left[\frac{15 k^{2}}{4(k|z|+1)^{2}}-3 k \delta(z)\right] \psi=0$$

  where $k$ is a positive constant and $\delta$ denotes the Dirac delta function. Physically relevant solutions for $\psi$ are bounded over the entire range $z \in \mathbb{R}$.

  (i) Find piecewise bounded solutions to this differential equations in the ranges $z>0$ and $z<0$, respectively. [Hint: The equation $\frac{d^{2} y}{d x^{2}}-\frac{c}{x^{2}} y=0$ for a constant $c$ may be solved using the Ansatz $y=x^{\alpha}$.]

  (ii) Derive a matching condition at $z=0$ by integrating ( $\dagger$ ) over the interval $(-\epsilon, \epsilon)$ with $\epsilon \rightarrow 0$ and use this condition together with the requirement that $\psi$ be continuous at $z=0$ to determine the solution over the entire range $z \in \mathbb{R}$.

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• Paper 2, Section I, D

  #Methods

  2014

  (i) Calculate the Fourier series for the periodic extension on $\mathbb{R}$ of the function

  $$f(x)=x(1-x)$$

  defined on the interval $[0,1)$.

  (ii) Explain why the Fourier series for the periodic extension of $f^{\prime}(x)$ can be obtained by term-by-term differentiation of the series for $f(x)$.

  (iii) Let $G(x)$ be the Fourier series for the periodic extension of $f^{\prime}(x)$. Determine the value of $G(0)$ and explain briefly how it is related to the values of $f^{\prime}$.

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• Paper 3, Section I, D

  #Methods

  2014

  Using the method of characteristics, solve the differential equation

  $$-y \frac{\partial u}{\partial x}+x \frac{\partial u}{\partial y}=0$$

  where $x, y \in \mathbb{R}$ and $u=\cos y^{2}$ on $x=0, y \geqslant 0$.

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• Paper 1, Section II, D

  #Methods

  2014

  (a) Legendre's differential equation may be written

  $$\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-2 x \frac{d y}{d x}+n(n+1) y=0, \quad y(1)=1$$

  Show that for non-negative integer $n$, this equation has a solution $P_{n}(x)$ that is a polynomial of degree $n$. Find $P_{0}, P_{1}$ and $P_{2}$ explicitly.

  (b) Laplace's equation in spherical coordinates for an axisymmetric function $U(r, \theta)$ (i.e. no $\phi$ dependence) is given by

  $$\frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial U}{\partial r}\right)+\frac{1}{r^{2} \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial U}{\partial \theta}\right)=0$$

  Use separation of variables to find the general solution for $U(r, \theta)$.

  Find the solution $U(r, \theta)$ that satisfies the boundary conditions

  $$\begin{aligned} &U(r, \theta) \rightarrow v_{0} r \cos \theta \quad \text { as } r \rightarrow \infty \\ &\frac{\partial U}{\partial r}=0 \quad \text { at } r=r_{0} \end{aligned}$$

  where $v_{0}$ and $r_{0}$ are constants.

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• Paper 3, Section II, 15D

  #Methods

  2014

  Let $\mathcal{L}$ be a linear second-order differential operator on the interval $[0, \pi / 2]$. Consider the problem

  $$\mathcal{L} y(x)=f(x) ; \quad y(0)=y(\pi / 2)=0$$

  with $f(x)$ bounded in $[0, \pi / 2]$.

  (i) How is a Green's function for this problem defined?

  (ii) How is a solution $y(x)$ for this problem constructed from the Green's function?

  (iii) Describe the continuity and jump conditions used in the construction of the Green's function.

  (iv) Use the continuity and jump conditions to construct the Green's function for the differential equation

  $$\frac{d^{2} y}{d x^{2}}-\frac{d y}{d x}+\frac{5}{4} y=f(x)$$

  on the interval $[0, \pi / 2]$ with the boundary conditions $y(0)=0, y(\pi / 2)=0$ and an arbitrary bounded function $f(x)$. Use the Green's function to construct a solution $y(x)$ for the particular case $f(x)=e^{x / 2}$.

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• Paper 2, Section II, 16D

  #Methods

  2014

  The Fourier transform $\tilde{f}$ of a function $f$ is defined as

  $$\tilde{f}(k)=\int_{-\infty}^{\infty} f(x) e^{-i k x} d x, \quad \text { so that } f(x)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} \tilde{f}(k) e^{i k x} d k$$

  A Green's function $G\left(t, t^{\prime}, x, x^{\prime}\right)$ for the diffusion equation in one spatial dimension satisfies

  $$\frac{\partial G}{\partial t}-D \frac{\partial^{2} G}{\partial x^{2}}=\delta\left(t-t^{\prime}\right) \delta\left(x-x^{\prime}\right)$$

  (a) By applying a Fourier transform, show that the Fourier transform $\tilde{G}$ of this Green's function and the Green's function $G$ are

  $$\begin{aligned} \tilde{G}\left(t, t^{\prime}, k, x^{\prime}\right) &=H\left(t-t^{\prime}\right) e^{-i k x^{\prime}} e^{-D k^{2}\left(t-t^{\prime}\right)} \\ G\left(t, t^{\prime}, x, x^{\prime}\right) &=\frac{H\left(t-t^{\prime}\right)}{\sqrt{4 \pi D\left(t-t^{\prime}\right)}} e^{-\frac{\left(x-x^{\prime}\right)^{2}}{4 D\left(t-t^{\prime}\right)}} \end{aligned}$$

  where $H$ is the Heaviside function. [Hint: The Fourier transform $\tilde{F}$ of a Gaussian $F(x)=\frac{1}{\sqrt{4 \pi a}} e^{-\frac{x^{2}}{4 a}}, a=\mathrm{const}$, is given by $\left.\tilde{F}(k)=e^{-a k^{2}} .\right]$

  (b) The analogous result for the Green's function for the diffusion equation in two spatial dimensions is

  $$G\left(t, t^{\prime}, x, x^{\prime}, y, y^{\prime}\right)=\frac{H\left(t-t^{\prime}\right)}{4 \pi D\left(t-t^{\prime}\right)} e^{-\frac{1}{4 D\left(t-t^{\prime}\right)}\left[\left(x-x^{\prime}\right)^{2}+\left(y-y^{\prime}\right)^{2}\right]}$$

  Use this Green's function to construct a solution for $t \geqslant 0$ to the diffusion equation

  $$\frac{\partial \Psi}{\partial t}-D\left(\frac{\partial^{2} \Psi}{\partial x^{2}}+\frac{\partial^{2} \Psi}{\partial y^{2}}\right)=p(t) \delta(x) \delta(y)$$

  with the initial condition $\Psi(0, x, y)=0$.

  Now set

  $$p(t)= \begin{cases}p_{0}=\mathrm{const} & \text { for } \quad 0 \leqslant t \leqslant t_{0} \\ 0 & \text { for } \quad t>t_{0}\end{cases}$$

  Find the solution $\Psi(t, x, y)$ for $t>t_{0}$ in terms of the exponential integral defined by

  $$E i(-\eta)=-\int_{\eta}^{\infty} \frac{e^{-\lambda}}{\lambda} d \lambda$$

  Use the approximation $E i(-\eta) \approx \ln \eta+C$, valid for $\eta \ll 1$, to simplify this solution $\Psi(t, x, y)$. Here $C \approx 0.577$ is Euler's constant.

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• Paper 4, Section II, D

  #Methods

  2014

  Let $f(x)$ be a complex-valued function defined on the interval $[-L, L]$ and periodically extended to $x \in \mathbb{R}$.

  (i) Express $f(x)$ as a complex Fourier series with coefficients $c_{n}, n \in \mathbb{Z}$. How are the coefficients $c_{n}$ obtained from $f(x)$ ?

  (ii) State Parseval's theorem for complex Fourier series.

  (iii) Consider the function $f(x)=\cos (\alpha x)$ on the interval $[-\pi, \pi]$ and periodically extended to $x \in \mathbb{R}$ for a complex but non-integer constant $\alpha$. Calculate the complex Fourier series of $f(x)$.

  (iv) Prove the formula

  $$\sum_{n=1}^{\infty} \frac{1}{n^{2}-\alpha^{2}}=\frac{1}{2 \alpha^{2}}-\frac{\pi}{2 \alpha \tan (\alpha \pi)}$$

  (v) Now consider the case where $\alpha$ is a real, non-integer constant. Use Parseval's theorem to obtain a formula for

  $$\sum_{n=-\infty}^{\infty} \frac{1}{\left(n^{2}-\alpha^{2}\right)^{2}}$$

  What value do you obtain for this series for $\alpha=5 / 2 ?$

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• Paper 3, Section I, E

  #Metric and Topological Spaces

  2014

  Suppose $(X, d)$ is a metric space. Do the following necessarily define a metric on $X$ ? Give proofs or counterexamples.

  (i) $d_{1}(x, y)=k d(x, y)$ for some constant $k>0$, for all $x, y \in X$.

  (ii) $d_{2}(x, y)=\min \{1, d(x, y)\}$ for all $x, y \in X$.

  (iii) $d_{3}(x, y)=(d(x, y))^{2}$ for all $x, y \in X$.

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• Paper 2, Section $I$, E

  #Metric and Topological Spaces

  2014

  Let $X$ and $Y$ be topological spaces. What does it mean to say that a function $f: X \rightarrow Y$ is continuous?

  Are the following statements true or false? Give proofs or counterexamples.

  (i) Every continuous function $f: X \rightarrow Y$ is an open map, i.e. if $U$ is open in $X$ then $f(U)$ is open in $Y$.

  (ii) If $f: X \rightarrow Y$ is continuous and bijective then $f$ is a homeomorphism.

  (iii) If $f: X \rightarrow Y$ is continuous, open and bijective then $f$ is a homeomorphism.

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• Paper 1, Section II, E

  #Metric and Topological Spaces

  2014

  Define what it means for a topological space to be compact. Define what it means for a topological space to be Hausdorff.

  Prove that a compact subspace of a Hausdorff space is closed. Hence prove that if $C_{1}$ and $C_{2}$ are compact subspaces of a Hausdorff space $X$ then $C_{1} \cap C_{2}$ is compact.

  A subset $U$ of $\mathbb{R}$ is open in the cocountable topology if $U$ is empty or its complement in $\mathbb{R}$ is countable. Is $\mathbb{R}$ Hausdorff in the cocountable topology? Which subsets of $\mathbb{R}$ are compact in the cocountable topology?

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• Paper 4, Section II, E

  #Metric and Topological Spaces

  2014

  Explain what it means for a metric space to be complete.

  Let $X$ be a metric space. We say the subsets $A_{i}$ of $X$, with $i \in \mathbb{N}$, form a descending sequence in $X$ if $A_{1} \supset A_{2} \supset A_{3} \supset \cdots$.

  Prove that the metric space $X$ is complete if and only if any descending sequence $A_{1} \supset A_{2} \supset \cdots$ of non-empty closed subsets of $X$, such that the diameters of the subsets $A_{i}$ converge to zero, has an intersection $\bigcap_{i=1}^{\infty} A_{i}$ that is non-empty.

  [Recall that the diameter $\operatorname{diam}(S)$ of a set $S$ is the supremum of the set $\{d(x, y)$ : $x, y \in S\} .]$

  Give examples of

  (i) a metric space $X$, and a descending sequence $A_{1} \supset A_{2} \supset \cdots$ of non-empty closed subsets of $X$, with $\operatorname{diam}\left(A_{i}\right)$ converging to 0 but $\bigcap_{i=1}^{\infty} A_{i}=\emptyset$.

  (ii) a descending sequence $A_{1} \supset A_{2} \supset \cdots$ of non-empty sets in $\mathbb{R}$ with $\operatorname{diam}\left(A_{i}\right)$ converging to 0 but $\bigcap_{i=1}^{\infty} A_{i}=\emptyset$.

  (iii) a descending sequence $A_{1} \supset A_{2} \supset \cdots$ of non-empty closed sets in $\mathbb{R}$ with $\bigcap_{i=1}^{\infty} A_{i}=\emptyset$.

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• Paper 1, Section I, $\mathbf{6 C}$

  #Numerical Analysis

  2014

  (i) A general multistep method for the numerical approximation to the scalar differential equation $y^{\prime}=f(t, y)$ is given by

  $$\sum_{\ell=0}^{s} \rho_{\ell} y_{n+\ell}=h \sum_{\ell=0}^{s} \sigma_{\ell} f_{n+\ell}, \quad n=0,1, \ldots$$

  where $f_{n+\ell}=f\left(t_{n+\ell}, y_{n+\ell}\right)$. Show that this method is of order $p \geqslant 1$ if and only if

  $$\rho\left(\mathrm{e}^{z}\right)-z \sigma\left(\mathrm{e}^{z}\right)=\mathcal{O}\left(z^{p+1}\right) \quad \text { as } \quad z \rightarrow 0$$

  where

  $$\rho(w)=\sum_{\ell=0}^{s} \rho_{\ell} w^{\ell} \quad \text { and } \quad \sigma(w)=\sum_{\ell=0}^{s} \sigma_{\ell} w^{\ell}$$

  (ii) A particular three-step implicit method is given by

  $$y_{n+3}+(a-1) y_{n+1}-a y_{n}=h\left(f_{n+3}+\sum_{\ell=0}^{2} \sigma_{\ell} f_{n+\ell}\right)$$

  where the $\sigma_{\ell}$ are chosen to make the method third order. [The $\sigma_{\ell}$ need not be found.] For what values of $a$ is the method convergent?

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• Paper 4, Section I, C

  #Numerical Analysis

  2014

  Consider the quadrature given by

  $$\int_{0}^{\pi} w(x) f(x) d x \approx \sum_{k=1}^{\nu} b_{k} f\left(c_{k}\right)$$

  for $\nu \in \mathbb{N}$, disjoint $c_{k} \in(0, \pi)$ and $w>0$. Show that it is not possible to make this quadrature exact for all polynomials of order $2 \nu$.

  For the case that $\nu=2$ and $w(x)=\sin x$, by considering orthogonal polynomials find suitable $b_{k}$ and $c_{k}$ that make the quadrature exact on cubic polynomials.

  [Hint: $\int_{0}^{\pi} x^{2} \sin x d x=\pi^{2}-4$ and $\int_{0}^{\pi} x^{3} \sin x d x=\pi^{3}-6 \pi .$ ]

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• Paper 1, Section II, C

  #Numerical Analysis

  2014

  Define a Householder transformation $\mathrm{H}$ and show that it is an orthogonal matrix. Briefly explain how these transformations can be used for QR factorisation of an $m \times n$ matrix.

  Using Householder transformations, find a QR factorisation of

  $$A=\left[\begin{array}{rrr} 2 & 5 & 4 \\ 2 & 5 & 1 \\ -2 & 1 & 5 \\ 2 & -1 & 16 \end{array}\right]$$

  Using this factorisation, find the value of $\lambda$ for which

  $$A x=\left[\begin{array}{c} 1+\lambda \\ 2 \\ 3 \\ 4 \end{array}\right]$$

  has a unique solution $x \in \mathbb{R}^{3}$.

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• Paper 3, Section II, C

  #Numerical Analysis

  2014

  A Runge-Kutta scheme is given by

  $$k_{1}=h f\left(y_{n}\right), \quad k_{2}=h f\left(y_{n}+\left[(1-a) k_{1}+a k_{2}\right]\right), \quad y_{n+1}=y_{n}+\frac{1}{2}\left(k_{1}+k_{2}\right)$$

  for the solution of an autonomous differential equation $y^{\prime}=f(y)$, where $a$ is a real parameter. What is the order of the scheme? Identify all values of $a$ for which the scheme is A-stable. Determine the linear stability domain for this range.

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• Paper 2, Section II, C

  #Numerical Analysis

  2014

  A linear functional acting on $f \in C^{k+1}[a, b]$ is approximated using a linear scheme $L(f)$. The approximation is exact when $f$ is a polynomial of degree $k$. The error is given by $\lambda(f)$. Starting from the Taylor formula for $f(x)$ with an integral remainder term, show that the error can be written in the form

  $$\lambda(f)=\frac{1}{k !} \int_{a}^{b} K(\theta) f^{(k+1)}(\theta) d \theta$$

  subject to a condition on $\lambda$ that you should specify. Give an expression for $K(\theta)$.

  Find $c_{0}, c_{1}$ and $c_{3}$ such that the approximation scheme

  $$f^{\prime \prime}(2) \approx c_{0} f(0)+c_{1} f(1)+c_{3} f(3)$$

  is exact for all $f$ that are polynomials of degree 2 . Assuming $f \in C^{3}[0,3]$, apply the Peano kernel theorem to the error. Find and sketch $K(\theta)$ for $k=2$.

  Find the minimum values for the constants $r$ and $s$ for which

  $$|\lambda(f)| \leqslant r\left\|f^{(3)}\right\|_{1} \quad \text { and } \quad|\lambda(f)| \leqslant s\left\|f^{(3)}\right\|_{\infty}$$

  and show explicitly that both error bounds hold for $f(x)=x^{3}$.

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• Paper 1, Section I, 8H

  #Optimization

  2014

  State and prove the Lagrangian sufficiency theorem.

  Use the Lagrangian sufficiency theorem to find the minimum of $2 x_{1}^{2}+2 x_{2}^{2}+x_{3}^{2}$ subject to $x_{1}+x_{2}+x_{3}=1$ (where $x_{1}, x_{2}$ and $x_{3}$ are real).

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• Paper 2, Section I, H

  #Optimization

  2014

  Explain what is meant by a two-player zero-sum game with $m \times n$ pay-off matrix $P=\left(p_{i j}\right)$, and state the optimal strategies for each player.

  Find these optimal strategies when

  $$P=\left(\begin{array}{rr} -4 & 2 \\ 2 & -4 \end{array}\right)$$

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• Paper 4, Section II, H

  #Optimization

  2014

  Consider a network with a single source and a single sink, where all the edge capacities are finite. Write down the maximum flow problem, and state the max-flow min-cut theorem.

  Describe the Ford-Fulkerson algorithm. If all edge capacities are integers, explain why, starting from a suitable initial flow, the algorithm is guaranteed to end after a finite number of iterations.

  The graph in the diagram below represents a one-way road network taking traffic from point $A$ to point $B$ via five roundabouts $R_{i}, i=1, \ldots, 5$. The capacity of each road is shown on the diagram in terms of vehicles per minute. Assuming that all roundabouts can deal with arbitrary amounts of flow of traffic, find the maximum flow of traffic (in vehicles per minute) through this network of roads. Show that this flow is indeed optimal.

  After a heavy storm, roundabout $R_{2}$ is flooded and only able to deal with at most 20 vehicles per minute. Find a suitable new network for the situation after the storm. Apply the Ford-Fulkerson algorithm to the new network, starting with the zero flow and explaining each step, to determine the maximum flow and the associated flows on each road.

  

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• Paper 3, Section II, H

  #Optimization

  2014

  Use the two-phase simplex method to maximise $2 x_{1}+x_{2}+x_{3}$ subject to the constraints

  $$x_{1}+x_{2} \geqslant 1, \quad x_{1}+x_{2}+2 x_{3} \leqslant 4, \quad x_{i} \geqslant 0 \text { for } i=1,2,3$$

  Derive the dual of this linear programming problem and find the optimal solution of the dual.

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• Paper 4, Section I, A

  #Quantum Mechanics

  2014

  For some quantum mechanical observable $Q$, prove that its uncertainty $(\Delta Q)$ satisfies

  $$(\Delta Q)^{2}=\left\langle Q^{2}\right\rangle-\langle Q\rangle^{2}$$

  A quantum mechanical harmonic oscillator has Hamiltonian

  $$H=\frac{p^{2}}{2 m}+\frac{m \omega^{2} x^{2}}{2}$$

  where $m>0$. Show that (in a stationary state of energy $E$ )

  $$E \geqslant \frac{(\Delta p)^{2}}{2 m}+\frac{m \omega^{2}(\Delta x)^{2}}{2}$$

  Write down the Heisenberg uncertainty relation. Then, use it to show that

  $$E \geqslant \frac{1}{2} \hbar \omega$$

  for our stationary state.

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• Paper 3, Section I, A

  #Quantum Mechanics

  2014

  The wavefunction of a normalised Gaussian wavepacket for a particle of mass $m$ in one dimension with potential $V(x)=0$ is given by

  $$\psi(x, t)=B \sqrt{A(t)} \exp \left(\frac{-x^{2} A(t)}{2}\right)$$

  where $A(0)=1$. Given that $\psi(x, t)$ is a solution of the time-dependent Schrödinger equation, find the complex-valued function $A(t)$ and the real constant $B$.

  [You may assume that $\int_{-\infty}^{\infty} e^{-\lambda x^{2}} d x=\sqrt{\pi} / \sqrt{\lambda} .$ ]

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• Paper 1, Section II, A

  #Quantum Mechanics

  2014

  Consider a particle confined in a one-dimensional infinite potential well: $V(x)=\infty$ for $|x| \geqslant a$ and $V(x)=0$ for $|x|<a$. The normalised stationary states are

  $$\psi_{n}(x)= \begin{cases}\alpha_{n} \sin \left(\frac{\pi n(x+a)}{2 a}\right) & \text { for }|x|<a \\ 0 & \text { for }|x| \geqslant a\end{cases}$$

  where $n=1,2, \ldots$.

  (i) Determine the $\alpha_{n}$ and the stationary states' energies $E_{n}$.

  (ii) A state is prepared within this potential well: $\psi(x) \propto x$ for $0<x<a$, but $\psi(x)=0$ for $x \leqslant 0$ or $x \geqslant a$. Find an explicit expansion of $\psi(x)$ in terms of $\psi_{n}(x) .$

  (iii) If the energy of the state is then immediately measured, show that the probability that it is greater than $\frac{\hbar^{2} \pi^{2}}{m a^{2}}$ is

  $$\sum_{n=0}^{4} \frac{b_{n}}{\pi^{n}}$$

  where the $b_{n}$ are integers which you should find.

  (iv) By considering the normalisation condition for $\psi(x)$ in terms of the expansion in $\psi_{n}(x)$, show that

  $$\frac{\pi^{2}}{3}=\sum_{p=1}^{\infty} \frac{A}{p^{2}}+\frac{B}{(2 p-1)^{2}}\left(1+\frac{C(-1)^{p}}{(2 p-1) \pi}\right)^{2}$$

  where $A, B$ and $C$ are integers which you should find.

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• Paper 3, Section II, A

  #Quantum Mechanics

  2014

  The Hamiltonian of a two-dimensional isotropic harmonic oscillator is given by

  $$H=\frac{p_{x}^{2}+p_{y}^{2}}{2 m}+\frac{m \omega^{2}}{2}\left(x^{2}+y^{2}\right)$$

  where $x$ and $y$ denote position operators and $p_{x}$ and $p_{y}$ the corresponding momentum operators.

  State without proof the commutation relations between the operators $x, y, p_{x}, p_{y}$. From these commutation relations, write $\left[x^{2}, p_{x}\right]$ and $\left[x, p_{x}^{2}\right]$ in terms of a single operator. Now consider the observable

  $$L=x p_{y}-y p_{x}$$

  Ehrenfest's theorem states that, for some observable $\mathrm{Q}$ with expectation value $\langle Q\rangle$,

  $$\frac{d\langle Q\rangle}{d t}=\frac{1}{i \hbar}\langle[Q, H]\rangle+\left\langle\frac{\partial Q}{\partial t}\right\rangle$$

  Use it to show that the expectation value of $L$ is constant with time.

  Given two states

  $$\psi_{1}=\alpha x \exp \left(-\beta\left(x^{2}+y^{2}\right)\right) \text { and } \psi_{2}=\alpha y \exp \left(-\beta\left(x^{2}+y^{2}\right)\right)$$

  where $\alpha$ and $\beta$ are constants, find a normalised linear combination of $\psi_{1}$ and $\psi_{2}$ that is an eigenstate of $L$, and the corresponding $L$ eigenvalue. [You may assume that $\alpha$ correctly normalises both $\psi_{1}$ and $\psi_{2}$.] If a quantum state is prepared in the linear combination you have found at time $t=0$, what is the expectation value of $L$ at a later time $t ?$

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• Paper 2, Section II, A

  #Quantum Mechanics

  2014

  For an electron of mass $m$ in a hydrogen atom, the time-independent Schrödinger equation may be written as

  $$-\frac{\hbar^{2}}{2 m r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial \psi}{\partial r}\right)+\frac{1}{2 m r^{2}} L^{2} \psi-\frac{e^{2}}{4 \pi \epsilon_{0} r} \psi=E \psi$$

  Consider normalised energy eigenstates of the form

  $$\psi_{l m}(r, \theta, \phi)=R(r) Y_{l m}(\theta, \phi)$$

  where $Y_{l m}$ are orbital angular momentum eigenstates:

  $$L^{2} Y_{l m}=\hbar^{2} l(l+1) Y_{l m}, \quad L_{3} Y_{l m}=\hbar m Y_{l m}$$

  where $l=1,2, \ldots$ and $m=0, \pm 1, \pm 2, \ldots \pm l$. The $Y_{l m}$ functions are normalised with $\int_{\theta=0}^{\pi} \int_{\phi=0}^{2 \pi}\left|Y_{l m}\right|^{2} \sin \theta d \theta d \phi=1 .$

  (i) Write down the resulting equation satisfied by $R(r)$, for fixed $l$. Show that it has solutions of the form

  $$R(r)=A r^{l} \exp \left(-\frac{r}{a(l+1)}\right)$$

  where $a$ is a constant which you should determine. Show that

  $$E=-\frac{e^{2}}{D \pi \epsilon_{0} a}$$

  where $D$ is an integer which you should find (in terms of $l$ ). Also, show that

  $$|A|^{2}=\frac{2^{2 l+3}}{a^{F} G !(l+1)^{H}},$$

  where $F, G$ and $H$ are integers that you should find in terms of $l$.

  (ii) Given the radius of the proton $r_{p} \ll a$, show that the probability of the electron being found within the proton is approximately

  $$\frac{2^{2 l+3}}{C}\left(\frac{r_{p}}{a}\right)^{2 l+3}\left[1+\mathcal{O}\left(\frac{r_{p}}{a}\right)\right]$$

  finding the integer $C$ in terms of $l$.

  [You may assume that $\int_{0}^{\infty} t^{l} e^{-t} d t=l !$.]

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• Paper 1, Section I, $\mathbf{7 H} \quad$

  #Statistics

  2014

  Consider an estimator $\hat{\theta}$ of an unknown parameter $\theta$, and assume that $\mathbb{E}_{\theta}\left(\hat{\theta}^{2}\right)<\infty$ for all $\theta$. Define the bias and mean squared error of $\hat{\theta}$.

  Show that the mean squared error of $\hat{\theta}$ is the sum of its variance and the square of its bias.

  Suppose that $X_{1}, \ldots, X_{n}$ are independent identically distributed random variables with mean $\theta$ and variance $\theta^{2}$, and consider estimators of $\theta$ of the form $k \bar{X}$ where $\bar{X}=\frac{1}{n} \sum_{i=1}^{n} X_{i}$.

  (i) Find the value of $k$ that gives an unbiased estimator, and show that the mean squared error of this unbiased estimator is $\theta^{2} / n$.

  (ii) Find the range of values of $k$ for which the mean squared error of $k \bar{X}$ is smaller $\operatorname{than} \theta^{2} / n$.

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• Paper 2, Section I, H

  #Statistics

  2014

  There are 100 patients taking part in a trial of a new surgical procedure for a particular medical condition. Of these, 50 patients are randomly selected to receive the new procedure and the remaining 50 receive the old procedure. Six months later, a doctor assesses whether or not each patient has fully recovered. The results are shown below:

  \begin{tabular}{l|c|c} & Fully recovered & Not fully recovered \ \hline Old procedure & 25 & 25 \ \hline New procedure & 31 & 19 \end{tabular}

  The doctor is interested in whether there is a difference in full recovery rates for patients receiving the two procedures. Carry out an appropriate $5 \%$ significance level test, stating your hypotheses carefully. [You do not need to derive the test.] What conclusion should be reported to the doctor?

  [Hint: Let $\chi_{k}^{2}(\alpha)$ denote the upper $100 \alpha$ percentage point of a $\chi_{k}^{2}$ distribution. Then

  $$\left.\chi_{1}^{2}(0.05)=3.84, \chi_{2}^{2}(0.05)=5.99, \chi_{3}^{2}(0.05)=7.82, \chi_{4}^{2}(0.05)=9.49 .\right]$$

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• Paper 4, Section II, H

  #Statistics

  2014

  Consider a linear model

  $$\mathbf{Y}=X \boldsymbol{\beta}+\varepsilon$$

  where $X$ is a known $n \times p$ matrix, $\boldsymbol{\beta}$ is a $p \times 1(p<n)$ vector of unknown parameters and $\varepsilon$ is an $n \times 1$ vector of independent $N\left(0, \sigma^{2}\right)$ random variables with $\sigma^{2}$ unknown. Assume that $X$ has full rank $p$. Find the least squares estimator $\hat{\boldsymbol{\beta}}$ of $\boldsymbol{\beta}$ and derive its distribution. Define the residual sum of squares $R S S$ and write down an unbiased estimator $\hat{\sigma}^{2}$ of $\sigma^{2}$.

  Suppose that $V_{i}=a+b u_{i}+\delta_{i}$ and $Z_{i}=c+d w_{i}+\eta_{i}$, for $i=1, \ldots, m$, where $u_{i}$ and $w_{i}$ are known with $\sum_{i=1}^{m} u_{i}=\sum_{i=1}^{m} w_{i}=0$, and $\delta_{1}, \ldots, \delta_{m}, \eta_{1}, \ldots, \eta_{m}$ are independent $N\left(0, \sigma^{2}\right)$ random variables. Assume that at least two of the $u_{i}$ are distinct and at least two of the $w_{i}$ are distinct. Show that $\mathbf{Y}=\left(V_{1}, \ldots, V_{m}, Z_{1}, \ldots, Z_{m}\right)^{T}$ (where $T$ denotes transpose) may be written as in ( $\dagger$ ) and identify $X$ and $\boldsymbol{\beta}$. Find $\hat{\boldsymbol{\beta}}$ in terms of the $V_{i}, Z_{i}$, $u_{i}$ and $w_{i}$. Find the distribution of $\hat{b}-\hat{d}$ and derive a $95 \%$ confidence interval for $b-d$.

  [Hint: You may assume that $\frac{R S S}{\sigma^{2}}$ has a $\chi_{n-p}^{2}$ distribution, and that $\hat{\beta}$ and the residual sum of squares are independent. Properties of $\chi^{2}$ distributions may be used without proof.]

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• Paper 1, Section II, H

  #Statistics

  2014

  Suppose that $X_{1}, X_{2}$, and $X_{3}$ are independent identically distributed Poisson random variables with expectation $\theta$, so that

  $$\mathbb{P}\left(X_{i}=x\right)=\frac{e^{-\theta} \theta^{x}}{x !} \quad x=0,1, \ldots$$

  and consider testing $H_{0}: \theta=1$ against $H_{1}: \theta=\theta_{1}$, where $\theta_{1}$ is a known value greater than 1. Show that the test with critical region $\left\{\left(x_{1}, x_{2}, x_{3}\right): \sum_{i=1}^{3} x_{i}>5\right\}$ is a likelihood ratio test of $H_{0}$ against $H_{1}$. What is the size of this test? Write down an expression for its power.

  A scientist counts the number of bird territories in $n$ randomly selected sections of a large park. Let $Y_{i}$ be the number of bird territories in the $i$ th section, and suppose that $Y_{1}, \ldots, Y_{n}$ are independent Poisson random variables with expectations $\theta_{1}, \ldots, \theta_{n}$ respectively. Let $a_{i}$ be the area of the $i$ th section. Suppose that $n=2 m$, $a_{1}=\cdots=a_{m}=a(>0)$ and $a_{m+1}=\cdots=a_{2 m}=2 a$. Derive the generalised likelihood ratio $\Lambda$ for testing

  $$H_{0}: \theta_{i}=\lambda a_{i} \text { against } H_{1}: \theta_{i}= \begin{cases}\lambda_{1} & i=1, \ldots, m \\ \lambda_{2} & i=m+1, \ldots, 2 m\end{cases}$$

  What should the scientist conclude about the number of bird territories if $2 \log _{e}(\Lambda)$ is $15.67 ?$

  [Hint: Let $F_{\theta}(x)$ be $\mathbb{P}(W \leqslant x)$ where $W$ has a Poisson distribution with expectation $\theta$. Then

  $$\left.F_{1}(3)=0.998, \quad F_{3}(5)=0.916, \quad F_{3}(6)=0.966, \quad F_{5}(3)=0.433 .\right]$$

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• Paper 3, Section II, H

  #Statistics

  2014

  Suppose that $X_{1}, \ldots, X_{n}$ are independent identically distributed random variables with

  $$\mathbb{P}\left(X_{i}=x\right)=\left(\begin{array}{c} k \\ x \end{array}\right) \theta^{x}(1-\theta)^{k-x}, \quad x=0, \ldots, k$$

  where $k$ is known and $\theta(0<\theta<1)$ is an unknown parameter. Find the maximum likelihood estimator $\hat{\theta}$ of $\theta$.

  Statistician 1 has prior density for $\theta$ given by $\pi_{1}(\theta)=\alpha \theta^{\alpha-1}, 0<\theta<1$, where $\alpha>1$. Find the posterior distribution for $\theta$ after observing data $X_{1}=x_{1}, \ldots, X_{n}=x_{n}$. Write down the posterior mean $\hat{\theta}_{1}^{(B)}$, and show that

  $$\hat{\theta}_{1}^{(B)}=c \hat{\theta}+(1-c) \tilde{\theta}_{1}$$

  where $\tilde{\theta}_{1}$ depends only on the prior distribution and $c$ is a constant in $(0,1)$ that is to be specified.

  Statistician 2 has prior density for $\theta$ given by $\pi_{2}(\theta)=\alpha(1-\theta)^{\alpha-1}, 0<\theta<1$. Briefly describe the prior beliefs that the two statisticians hold about $\theta$. Find the posterior mean $\hat{\theta}_{2}^{(B)}$ and show that $\hat{\theta}_{2}^{(B)}<\hat{\theta}_{1}^{(B)}$.

  Suppose that $\alpha$ increases (but $n, k$ and the $x_{i}$ remain unchanged). How do the prior beliefs of the two statisticians change? How does $c$ vary? Explain briefly what happens to $\hat{\theta}_{1}^{(B)}$ and $\hat{\theta}_{2}^{(B)}$.

  [Hint: The Beta $(\alpha, \beta)(\alpha>0, \beta>0)$ distribution has density

  $$f(x)=\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha) \Gamma(\beta)} x^{\alpha-1}(1-x)^{\beta-1}, \quad 0<x<1$$

  with expectation $\frac{\alpha}{\alpha+\beta}$ and variance $\frac{\alpha \beta}{(\alpha+\beta+1)(\alpha+\beta)^{2}}$. Here, $\Gamma(\alpha)=\int_{0}^{\infty} x^{\alpha-1} e^{-x} d x, \alpha>0$, is the Gamma function.]

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• Paper 1, Section I, C

  #Variational Principles

  2014

  Define the Legendre transform $f^{*}(\mathbf{p})$ of a function $f(\mathbf{x})$ where $\mathbf{x} \in \mathbb{R}^{n}$.

  Show that for $g(\mathbf{x})=\lambda f\left(\mathbf{x}-\mathbf{x}_{0}\right)-\mu$,

  $$g^{*}(\mathbf{p})=\lambda f^{*}\left(\frac{\mathbf{p}}{\lambda}\right)+\mathbf{p}^{\mathbf{T}} \mathbf{x}_{0}+\mu$$

  Show that for $f(\mathbf{x})=\frac{1}{2} \mathbf{x}^{\mathbf{T}} \mathbf{A} \mathbf{x}$ where $\mathbf{A}$ is a real, symmetric, invertible matrix with positive eigenvalues,

  $$f^{*}(\mathbf{p})=\frac{1}{2} \mathbf{p}^{\mathbf{T}} \mathbf{A}^{-\mathbf{1}} \mathbf{p}$$

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• Paper 3, Section I, $\mathbf{6 C}$

  #Variational Principles

  2014

  Let $f(x, y, z)=x z+y z$. Using Lagrange multipliers, find the location(s) and value of the maximum of $f$ on the intersection of the unit sphere $\left(x^{2}+y^{2}+z^{2}=1\right)$ and the ellipsoid given by $\frac{1}{4} x^{2}+\frac{1}{4} y^{2}+4 z^{2}=1$.

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• Paper 4, Section II, H

  #Algebraic Geometry

  2014

  Let $X$ be a smooth projective curve of genus $g>0$ over an algebraically closed field of characteristic $\neq 2$, and suppose there is a degree 2 morphism $\pi: X \rightarrow \mathbf{P}^{1}$. How many ramification points of $\pi$ are there?

  Suppose $Q$ and $R$ are distinct ramification points of $\pi$. Show that $Q \nsim R$, but $2 Q \sim 2 R$.

  Now suppose $g=2$. Show that every divisor of degree 2 on $X$ is linearly equivalent to $P+P^{\prime}$ for some $P, P^{\prime} \in X$, and deduce that every divisor of degree 0 is linearly equivalent to $P_{1}-P_{2}$ for some $P_{1}, P_{2} \in X$.

  Show that the subgroup $\left\{[D] \in C l^{0}(X) \mid 2[D]=0\right\}$ of the divisor class group of $X$ has order $16 .$

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• Paper 3, Section II, H

  #Algebraic Geometry

  2014

  Let $f \in k[x]$ be a polynomial with distinct roots, $\operatorname{deg} f=d>2$, char $k=0$, and let $C \subseteq \mathbf{P}^{2}$ be the projective closure of the affine curve

  $$y^{d-1}=f(x)$$

  Show that $C$ is smooth, with a single point at $\infty$.

  Pick an appropriate $\omega \in \Omega_{k(C) / k}^{1}$ and compute the valuation $v_{q}(\omega)$ for all $q \in C$.

  Hence determine $\operatorname{deg} \mathcal{K}_{C}$.

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• Paper 2, Section II, H

  #Algebraic Geometry

  2014

  (i) Let $k$ be an algebraically closed field, $n \geqslant 1$, and $S$ a subset of $k^{n}$.

  Let $I(S)=\left\{f \in k\left[x_{1}, \ldots, x_{n}\right] \mid f(p)=0\right.$ when $\left.p \in S\right\}$. Show that $I(S)$ is an ideal, and that $k\left[x_{1}, \ldots, x_{n}\right] / I(S)$ does not have any non-zero nilpotent elements.

  Let $X \subseteq \mathbf{A}^{n}, Y \subseteq \mathbf{A}^{m}$ be affine varieties, and $\Phi: k[Y] \rightarrow k[X]$ be a $k$-algebra homomorphism. Show that $\Phi$ determines a map of sets from $X$ to $Y$.

  (ii) Let $X$ be an irreducible affine variety. Define the dimension of $X, \operatorname{dim} X$ (in terms of the tangent spaces of $X$ ) and the transcendence dimension of $X, \operatorname{tr} \cdot \operatorname{dim} X$.

  State the Noether normalization theorem. Using this, or otherwise, prove that the transcendence dimension of $X$ equals the dimension of $X$.

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• Paper 1, Section II, H

  #Algebraic Geometry

  2014

  Let $k$ be an algebraically closed field and $n \geqslant 1$. We say that $f \in k\left[x_{1}, \ldots, x_{n}\right]$ is singular at $p \in \mathbf{A}^{n}$ if either $p$ is a singularity of the hypersurface $\{f=0\}$ or $f$ has an irreducible factor $h$ of multiplicity strictly greater than one with $h(p)=0$. Given $d \geqslant 1$, let $X=\left\{f \in k\left[x_{1}, \ldots, x_{n}\right] \mid \operatorname{deg} f \leqslant d\right\}$ and let

  $$Y=\left\{(f, p) \in X \times \mathbf{A}^{n} \mid f \text { is singular at } p\right\}$$

  (i) Show that $X \simeq \mathbf{A}^{N}$ for some $N$ (you need not determine $N$ ) and that $Y$ is a Zariski closed subvariety of $X \times \mathbf{A}^{n}$.

  (ii) Show that the fibres of the projection map $Y \rightarrow \mathbf{A}^{n}$ are linear subspaces of $\operatorname{dim} e n s i o n N-(n+1)$. Conclude that $\operatorname{dim} Y<\operatorname{dim} X$.

  (iii) Hence show that $\{f \in X \mid \operatorname{deg} f=d, Z(f)$ smooth $\}$ is dense in $X$.

  [You may use standard results from lectures if they are accurately quoted.]

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• Paper 3, Section II, F

  #Algebraic Topology

  2014

  Let $K$ be a simplicial complex in $\mathbb{R}^{N}$, which we may also consider as lying in $\mathbb{R}^{N+1}$ using the first $N$ coordinates. Write $c=(0,0, \ldots, 0,1) \in \mathbb{R}^{N+1}$. Show that if $\left\langle v_{0}, v_{1}, \ldots, v_{n}\right\rangle$ is a simplex of $K$ then $\left\langle v_{0}, v_{1}, \ldots, v_{n}, c\right\rangle$ is a simplex in $\mathbb{R}^{N+1}$.

  Let $L \leqslant K$ be a subcomplex and let $\bar{K}$ be the collection

  $$K \cup\left\{\left\langle v_{0}, v_{1}, \ldots, v_{n}, c\right\rangle \mid\left\langle v_{0}, v_{1}, \ldots, v_{n}\right\rangle \in L\right\} \cup\{\langle c\rangle\}$$

  of simplices in $\mathbb{R}^{N+1}$. Show that $\bar{K}$ is a simplicial complex.

  If $|K|$ is a Möbius band, and $|L|$ is its boundary, show that

  $$H_{i}(\bar{K}) \cong \begin{cases}\mathbb{Z} & \text { if } i=0 \\ \mathbb{Z} / 2 & \text { if } i=1 \\ 0 & \text { if } i \geqslant 2\end{cases}$$

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• Paper 4, Section II, F

  #Algebraic Topology

  2014

  State the Lefschetz fixed point theorem.

  Let $X$ be an orientable surface of genus $g$ (which you may suppose has a triangulation), and let $f: X \rightarrow X$ be a continuous map such that

  1. $f^{3}=\operatorname{Id}_{X}$,

  2. $f$ has no fixed points.

  By considering the eigenvalues of the linear map $f_{*}: H_{1}(X ; \mathbb{Q}) \rightarrow H_{1}(X ; \mathbb{Q})$, and their multiplicities, show that $g$ must be congruent to 1 modulo 3 .

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• Paper 2, Section II, F

  #Algebraic Topology

  2014

  Let $A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$ be a matrix with integer entries. Considering $S^{1}$ as the quotient space $\mathbb{R} / \mathbb{Z}$, show that the function

  $$\begin{aligned} \varphi_{A}: S^{1} \times S^{1} & \longrightarrow S^{1} \times S^{1} \\ ([x],[y]) & \longmapsto([a x+b y],[c x+d y]) \end{aligned}$$

  is well-defined and continuous. If in addition $\operatorname{det}(A)=\pm 1$, show that $\varphi_{A}$ is a homeomorphism.

  State the Seifert-van Kampen theorem. Let $X_{A}$ be the space obtained by gluing together two copies of $S^{1} \times D^{2}$ along their boundaries using the homeomorphism $\varphi_{A}$. Show that the fundamental group of $X_{A}$ is cyclic and determine its order.

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• Paper 1, Section II, F

  #Algebraic Topology

  2014

  Define what it means for a map $p: \widetilde{X} \rightarrow X$ to be a covering space. State the homotopy lifting lemma.

  Let $p:\left(\tilde{X}, \tilde{x}_{0}\right) \rightarrow\left(X, x_{0}\right)$ be a based covering space and let $f:\left(Y, y_{0}\right) \rightarrow\left(X, x_{0}\right)$ be a based map from a path-connected and locally path-connected space. Show that there is a based lift $\tilde{f}:\left(Y, y_{0}\right) \rightarrow\left(\tilde{X}, \tilde{x}_{0}\right)$ of $f$ if and only if $f_{*}\left(\pi_{1}\left(Y, y_{0}\right)\right) \subseteq p_{*}\left(\pi_{1}\left(\widetilde{X}, \tilde{x}_{0}\right)\right)$.

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• Paper 4, Section II, A

  #Applications of Quantum Mechanics

  2014

  Let $\Lambda$ be a Bravais lattice in three dimensions. Define the reciprocal lattice $\Lambda^{*}$.

  State and prove Bloch's theorem for a particle moving in a potential $V(\mathbf{x})$ obeying

  $$V(\mathbf{x}+\ell)=V(\mathbf{x}) \quad \forall \ell \in \Lambda, \mathbf{x} \in \mathbb{R}^{3}$$

  Explain what is meant by a Brillouin zone for this potential and how it is related to the reciprocal lattice.

  A simple cubic lattice $\Lambda_{1}$ is given by the set of points

  $$\Lambda_{1}=\left\{\ell \in \mathbb{R}^{3}: \ell=n_{1} \hat{\mathbf{i}}+n_{2} \hat{\mathbf{j}}+n_{3} \hat{\mathbf{k}}, n_{1}, n_{2}, n_{3} \in \mathbb{Z}\right\}$$

  where $\hat{\mathbf{i}}, \hat{\mathbf{j}}$ and $\hat{\mathbf{k}}$ are unit vectors parallel to the Cartesian coordinate axes in $\mathbb{R}^{3}$. A bodycentred cubic $\left(\mathrm{BCC}\right.$ ) lattice $\Lambda_{B C C}$ is obtained by adding to $\Lambda_{1}$ the points at the centre of each cube, i.e. all points of the form

  $$\ell+\frac{1}{2}(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}), \quad \ell \in \Lambda_{1}$$

  Show that $\Lambda_{B C C}$ is Bravais with primitive vectors

  $$\begin{aligned} \mathbf{a}_{1} &=\frac{1}{2}(\hat{\mathbf{j}}+\hat{\mathbf{k}}-\hat{\mathbf{i}}) \\ \mathbf{a}_{2} &=\frac{1}{2}(\hat{\mathbf{k}}+\hat{\mathbf{i}}-\hat{\mathbf{j}}) \\ \mathbf{a}_{3} &=\frac{1}{2}(\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}) \end{aligned}$$

  Find the reciprocal lattice $\Lambda_{B C C}^{*}$. Hence find a consistent choice for the first Brillouin zone of a potential $V(\mathbf{x})$ obeying

  $$\begin{aligned} & V(\mathbf{x}+\ell)=V(\mathbf{x}) \quad \forall \ell \in \Lambda_{B C C}, \mathbf{x} \in \mathbb{R}^{3} \\ & \text { [Hint: The matrix } \left.M=\frac{1}{2}\left(\begin{array}{rrr}-1 & 1 & 1 \\1 & -1 & 1 \\1 & 1 & -1\end{array}\right) \text { has inverse } M^{-1}=\left(\begin{array}{lll}0 & 1 & 1 \\1 & 0 & 1 \\1 & 1 & 0\end{array}\right) .\right] \end{aligned}$$

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• Paper 2, Section II, A

  #Applications of Quantum Mechanics

  2014

  (a) A classical particle of mass $m$ scatters on a central potential $V(r)$ with energy $E$, impact parameter $b$, and scattering angle $\theta$. Define the corresponding differential cross-section.

  For particle trajectories in the Coulomb potential,

  $$V_{C}(r)=\frac{e^{2}}{4 \pi \epsilon_{0} r}$$

  the impact parameter is given by

  $$b=\frac{e^{2}}{8 \pi \epsilon_{0} E} \cot \left(\frac{\theta}{2}\right)$$

  Find the differential cross-section as a function of $E$ and $\theta$.

  (b) A quantum particle of mass $m$ and energy $E=\hbar^{2} k^{2} / 2 m$ scatters in a localised potential $V(\mathbf{r})$. With reference to the asymptotic form of the wavefunction at large $|\mathbf{r}|$, define the scattering amplitude $f\left(\mathbf{k}, \mathbf{k}^{\prime}\right)$ as a function of the incident and outgoing wavevectors $\mathbf{k}$ and $\mathbf{k}^{\prime}$ (where $|\mathbf{k}|=\left|\mathbf{k}^{\prime}\right|=k$ ). Define the differential cross-section for this process and express it in terms of $f\left(\mathbf{k}, \mathbf{k}^{\prime}\right)$.

  Now consider a potential of the form $V(\mathbf{r})=\lambda U(\mathbf{r})$, where $\lambda \ll 1$ is a dimensionless coupling and $U$ does not depend on $\lambda$. You may assume that the Schrödinger equation for the wavefunction $\psi(\mathbf{k} ; \mathbf{r})$ of a scattering state with incident wavevector $\mathbf{k}$ may be written as the integral equation

  $$\psi(\mathbf{k} ; \mathbf{r})=\exp (i \mathbf{k} \cdot \mathbf{r})+\frac{2 m \lambda}{\hbar^{2}} \int d^{3} r^{\prime} \mathcal{G}_{0}^{(+)}\left(k ; \mathbf{r}-\mathbf{r}^{\prime}\right) U\left(\mathbf{r}^{\prime}\right) \psi\left(\mathbf{k} ; \mathbf{r}^{\prime}\right)$$

  where

  $$\mathcal{G}_{0}^{(+)}(k ; \mathbf{r})=-\frac{1}{4 \pi} \frac{\exp (i k|\mathbf{r}|)}{|\mathbf{r}|}$$

  Show that the corresponding scattering amplitude is given by

  $$f\left(\mathbf{k}, \mathbf{k}^{\prime}\right)=-\frac{m \lambda}{2 \pi \hbar^{2}} \int d^{3} r^{\prime} \exp \left(-i \mathbf{k}^{\prime} \cdot \mathbf{r}^{\prime}\right) U\left(\mathbf{r}^{\prime}\right) \psi\left(\mathbf{k} ; \mathbf{r}^{\prime}\right)$$

  By expanding the wavefunction in powers of $\lambda$ and keeping only the leading term, calculate the leading-order contribution to the differential cross-section, and evaluate it for the case of the Yukawa potential

  $$V(\mathbf{r})=\lambda \frac{\exp (-\mu r)}{r}$$

  By taking a suitable limit, obtain the differential cross-section for quantum scattering in the Coulomb potential $V_{C}(r)$ defined in Part (a) above, correct to leading order in an expansion in powers of the constant $\tilde{\alpha}=e^{2} / 4 \pi \epsilon_{0}$. Express your answer as a function of the particle energy $E$ and scattering angle $\theta$, and compare it to the corresponding classical cross-section calculated in Part (a).

  Part II, $2014 \quad$ List of Questions

  [TURN OVER

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• Paper 3, Section II, A

  #Applications of Quantum Mechanics

  2014

  In the nearly-free electron model a particle of mass $m$ moves in one dimension in a periodic potential of the form $V(x)=\lambda U(x)$, where $\lambda \ll 1$ is a dimensionless coupling and $U(x)$ has a Fourier series

  $$U(x)=\sum_{l=-\infty}^{+\infty} U_{l} \exp \left(\frac{2 \pi i}{a} l x\right)$$

  with coefficients obeying $U_{-l}=U_{l}^{*}$ for all $l$.

  Ignoring any degeneracies in the spectrum, the exact energy $E(k)$ of a Bloch state with wavenumber $k$ can be expanded in powers of $\lambda$ as

  $$E(k)=E_{0}(k)+\lambda\langle k|U| k\rangle+\lambda^{2} \sum_{k^{\prime} \neq k} \frac{\left\langle k|U| k^{\prime}\right\rangle\left\langle k^{\prime}|U| k\right\rangle}{E_{0}(k)-E_{0}\left(k^{\prime}\right)}+O\left(\lambda^{3}\right)$$

  where $|k\rangle$ is a normalised eigenstate of the free Hamiltonian $\hat{H}_{0}=\hat{p}^{2} / 2 m$ with momentum $p=\hbar k$ and energy $E_{0}(k)=\hbar^{2} k^{2} / 2 m$.

  Working on a finite interval of length $L=N a$, where $N$ is a positive integer, we impose periodic boundary conditions on the wavefunction:

  $$\psi(x+N a)=\psi(x)$$

  What are the allowed values of the wavenumbers $k$ and $k^{\prime}$ which appear in (1)? For these values evaluate the matrix element $\left\langle k|U| k^{\prime}\right\rangle$.

  For what values of $k$ and $k^{\prime}$ does (1) cease to be a good approximation? Explain your answer. Quoting any results you need from degenerate perturbation theory, calculate to $O(\lambda)$ the location and width of the gaps between allowed energy bands for the periodic potential $V(x)$, in terms of the Fourier coefficients $U_{l}$.

  Hence work out the allowed energy bands for the following potentials:

  $$\begin{aligned} &\text { (i) } \quad V(x)=2 \lambda \cos \left(\frac{2 \pi x}{a}\right), \\ &\text { (ii) } \quad V(x)=\lambda a \sum_{n=-\infty}^{+\infty} \delta(x-n a) . \end{aligned}$$

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• Paper 1, Section II, A

  #Applications of Quantum Mechanics

  2014

  A particle of mass $m$ scatters on a localised potential well $V(x)$ in one dimension. With reference to the asymptotic behaviour of the wavefunction as $x \rightarrow \pm \infty$, define the reflection and transmission amplitudes, $r$ and $t$, for a right-moving incident particle of wave number $k$. Define also the corresponding amplitudes, $r^{\prime}$ and $t^{\prime}$, for a left-moving incident particle of wave number $k$. Derive expressions for $r^{\prime}$ and $t^{\prime}$ in terms of $r$ and $t$.

  (a) Define the $S$-matrix, giving its elements in terms of $r$ and $t$. Using the relation

  $$|r|^{2}+|t|^{2}=1$$

  (which you need not derive), show that the S-matrix is unitary. How does the S-matrix simplify if the potential well satisfies $V(-x)=V(x)$ ?

  (b) Consider the potential well

  $$V(x)=-\frac{3 \hbar^{2}}{m} \frac{1}{\cosh ^{2}(x)}$$

  The corresponding Schrödinger equation has an exact solution

  $$\psi_{k}(x)=\exp (i k x)\left[3 \tanh ^{2}(x)-3 i k \tanh (x)-\left(1+k^{2}\right)\right]$$

  with energy $E=\hbar^{2} k^{2} / 2 m$, for every real value of $k$. [You do not need to verify this.] Find the S-matrix for scattering on this potential. What special feature does the scattering have in this case?

  (c) Explain the connection between singularities of the S-matrix and bound states of the potential well. By analytic continuation of the solution $\psi_{k}(x)$ to appropriate complex values of $k$, find the wavefunctions and energies of the bound states of the well. [You do not need to normalise the wavefunctions.]

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• Paper 3, Section II, J

  #Applied Probability

  2014

  (i) Define a Poisson process $\left(N_{t}, t \geqslant 0\right)$ with intensity $\lambda$. Specify without justification the distribution of $N_{t}$. Let $T_{1}, T_{2}, \ldots$ denote the jump times of $\left(N_{t}, t \geqslant 0\right)$. Derive the joint distribution of $\left(T_{1}, \ldots, T_{n}\right)$ given $\left\{N_{t}=n\right\}$.

  (ii) Let $\left(N_{t}, t \geqslant 0\right)$ be a Poisson process with intensity $\lambda>0$ and let $X_{1}, X_{2}, \ldots$ be a sequence of i.i.d. random variables, independent of $\left(N_{t}, t \geqslant 0\right)$, all having the same distribution as a random variable $X$. Show that if $g(s, x)$ is a real-valued function of real variables $s, x$, and $T_{j}$ are the jump times of $\left(N_{t}, t \geqslant 0\right)$ then

  $$\mathbb{E}\left[\exp \left\{\theta \sum_{j=1}^{N_{t}} g\left(T_{j}, X_{j}\right)\right\}\right]=\exp \left\{\lambda \int_{0}^{t}\left(\mathbb{E}\left(e^{\theta g(s, X)}\right)-1\right) d s\right\}$$

  for all $\theta \in \mathbb{R}$. [Hint: Condition on $\left\{N_{t}=n\right\}$ and $T_{1}, \ldots, T_{n}$, using (i).]

  (iii) A university library is open from 9 am to $5 \mathrm{pm}$. Students arrive at times of a Poisson process with intensity $\lambda$. Each student spends a random amount of time in the library, independently of the other students. These times are identically distributed for all students and have the same distribution as a random variable $X$. Show that the number of students in the library at $5 \mathrm{pm}$ is a Poisson random variable with a mean that you should specify.

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• Paper 4, Section II, J

  #Applied Probability

  2014

  (i) Define the $M / M / 1$ queue with arrival rate $\lambda$ and service rate $\mu$. Find conditions on the parameters $\lambda$ and $\mu$ for the queue to be transient, null recurrent, and positive recurrent, briefly justifying your answers. In the last case give with justification the invariant distribution explicitly. Answer the same questions for an $M / M / \infty$ queue.

  (ii) At a taxi station, customers arrive at a rate of 3 per minute, and taxis at a rate of 2 per minute. Suppose that a taxi will wait no matter how many other taxis are present. However, if a person arriving does not find a taxi waiting he or she leaves to find alternative transportation.

  Find the long-run proportion of arriving customers who get taxis, and find the average number of taxis waiting in the long run.

  An agent helps to assign customers to taxis, and so long as there are taxis waiting he is unable to have his coffee. Once a taxi arrives, how long will it take on average before he can have another sip of his coffee?

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• Paper 1, Section II, J

  #Applied Probability

  2014

  (i) Explain what a $Q$-matrix is. Let $Q$ be a $Q$-matrix. Define the notion of a Markov chain $\left(X_{t}, t \geqslant 0\right)$ in continuous time with $Q$-matrix given by $Q$, and give a construction of $\left(X_{t}, t \geqslant 0\right)$. [You are not required to justify this construction.]

  (ii) A population consists of $N_{t}$ individuals at time $t \geqslant 0$. We assume that each individual gives birth to a new individual at constant rate $\lambda>0$. As the population is competing for resources, we assume that for each $n \geqslant 1$, if $N_{t}=n$, then any individual in the population at time $t$ dies in the time interval $[t, t+h)$ with probability $\delta_{n} h+o(h)$, where $\left(\delta_{n}\right)_{n=1}^{\infty}$ is a given sequence satisfying $\delta_{1}=0, \delta_{n}>0$ for $n \geqslant 2$. Formulate a Markov chain model for $\left(N_{t}, t \geqslant 0\right)$ and write down the $Q$-matrix explicitly. Then find a necessary and sufficient condition on $\left(\delta_{n}\right)_{n=1}^{\infty}$ so that the Markov chain has an invariant distribution. Compute the invariant distribution in the case where $\delta_{n}=\mu(n-1)$ and $\mu>0$.

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• Paper 2, Section II, J

  #Applied Probability

  2014

  (i) Explain what the Moran model and the infinite alleles model are. State Ewens' sampling formula for the distribution of the allelic frequency spectrum $\left(a_{1}, \ldots, a_{n}\right)$ in terms of $\theta$ where $\theta=N u$ with $u$ denoting the mutation rate per individual and $N$ the population size.

  Let $K_{n}$ be the number of allelic types in a sample of size $n$. Give, without justification, an expression for $\mathbb{E}\left(K_{n}\right)$ in terms of $\theta$.

  (ii) Let $K_{n}$ and $\theta$ be as above. Show that for $1 \leqslant k \leqslant n$ we have that

  $$P\left(K_{n}=k\right)=C \frac{\theta^{k}}{\theta(\theta+1) \cdots(\theta+n-1)}$$

  for some constant $C$ that does not depend on $\theta$.

  Show that, given $\left\{K_{n}=k\right\}$, the distribution of the allelic frequency spectrum $\left(a_{1}, \ldots, a_{n}\right)$ does not depend on $\theta$.

  Show that the value of $\theta$ which maximises $\mathbb{P}\left(K_{n}=k\right)$ is the one for which $k=\mathbb{E}\left(K_{n}\right)$.

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• Paper 4, Section II, C

  #Asymptotic Methods

  2014

  Derive the leading-order Liouville Green (or WKBJ) solution for $\epsilon \ll 1$ to the ordinary differential equation

  $$\epsilon^{2} \frac{d^{2} f}{d y^{2}}+\Phi(y) f=0$$

  where $\Phi(y)>0$.

  The function $f(y ; \epsilon)$ satisfies the ordinary differential equation

  $$\epsilon^{2} \frac{d^{2} f}{d y^{2}}+\left(1+\frac{1}{y}-\frac{2 \epsilon^{2}}{y^{2}}\right) f=0$$

  subject to the boundary condition $f^{\prime \prime}(0)=2$. Show that the Liouville-Green solution of (1) for $\epsilon \ll 1$ takes the asymptotic forms

  where $\alpha_{1}, \alpha_{2}, B$ and $\theta_{2}$ are constants.

  $\left[\right.$ Hint: You may assume that $\left.\int_{0}^{y} \sqrt{1+u^{-1}} d u=\sqrt{y(1+y)}+\sinh ^{-1} \sqrt{y} \cdot\right]$

  Explain, showing the relevant change of variables, why the leading-order asymptotic behaviour for $0 \leqslant y \ll 1$ can be obtained from the reduced equation

  $$\frac{d^{2} f}{d x^{2}}+\left(\frac{1}{x}-\frac{2}{x^{2}}\right) f=0$$

  The unique solution to $(2)$ with $f^{\prime \prime}(0)=2$ is $f=x^{1 / 2} J_{3}\left(2 x^{1 / 2}\right)$, where the Bessel function $J_{3}(z)$ is known to have the asymptotic form

  $$J_{3}(z) \sim\left(\frac{2}{\pi z}\right)^{1 / 2} \cos \left(z-\frac{7 \pi}{4}\right) \text { as } z \rightarrow \infty .$$

  Hence find the values of $\alpha_{1}$ and $\alpha_{2}$.

  $$\begin{aligned} & f \sim \alpha_{1} y^{\frac{1}{4}} \exp (2 i \sqrt{y} / \epsilon)+\alpha_{2} y^{\frac{1}{4}} \exp (-2 i \sqrt{y} / \epsilon) \quad \text { for } \quad \epsilon^{2} \ll y \ll 1 \\ & \text { and } \quad f \sim B \cos \left[\theta_{2}+(y+\log \sqrt{y}) / \epsilon\right] \quad \text { for } \quad y \gg 1, \end{aligned}$$

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• Paper 3, Section II, C

  #Asymptotic Methods

  2014

  (a) Find the Stokes ray for the function $f(z)$ as $z \rightarrow 0$ with $0<\arg z<\pi$, where

  $$f(z)=\sinh \left(z^{-1}\right)$$

  (b) Describe how the leading-order asymptotic behaviour as $x \rightarrow \infty$ of

  $$I(x)=\int_{a}^{b} f(t) e^{i x g(t)} d t$$

  may be found by the method of stationary phase, where $f$ and $g$ are real functions and the integral is taken along the real line. You should consider the cases for which:

  (i) $g^{\prime}(t)$ is non-zero in $[a, b)$ and has a simple zero at $t=b$.

  (ii) $g^{\prime}(t)$ is non-zero apart from having one simple zero at $t=t_{0}$, where $a<t_{0}<b$.

  (iii) $g^{\prime}(t)$ has more than one simple zero in $(a, b)$ with $g^{\prime}(a) \neq 0$ and $g^{\prime}(b) \neq 0$.

  Use the method of stationary phase to find the leading-order asymptotic form as $x \rightarrow \infty$ of

  $$J(x)=\int_{0}^{1} \cos \left(x\left(t^{4}-t^{2}\right)\right) d t$$

  [You may assume that $\left.\int_{-\infty}^{\infty} e^{i u^{2}} d u=\sqrt{\pi} e^{i \pi / 4} .\right]$

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• Paper 1, Section II, C

  #Asymptotic Methods

  2014

  (a) Consider the integral

  $$I(k)=\int_{0}^{\infty} f(t) e^{-k t} d t, \quad k>0$$

  Suppose that $f(t)$ possesses an asymptotic expansion for $t \rightarrow 0^{+}$of the form

  $$f(t) \sim t^{\alpha} \sum_{n=0}^{\infty} a_{n} t^{\beta n}, \quad \alpha>-1, \quad \beta>0$$

  where $a_{n}$ are constants. Derive an asymptotic expansion for $I(k)$ as $k \rightarrow \infty$ in the form

  $$I(k) \sim \sum_{n=0}^{\infty} \frac{A_{n}}{k^{\gamma+\beta n}}$$

  giving expressions for $A_{n}$ and $\gamma$ in terms of $\alpha, \beta, n$ and the gamma function. Hence establish the asymptotic approximation as $k \rightarrow \infty$

  $$I_{1}(k)=\int_{0}^{1} e^{k t} t^{-a}\left(1-t^{2}\right)^{-b} d t \sim 2^{-b} \Gamma(1-b) e^{k} k^{b-1}\left(1+\frac{(a+b / 2)(1-b)}{k}\right)$$

  where $a<1, b<1$.

  (b) Using Laplace's method, or otherwise, find the leading-order asymptotic approximation as $k \rightarrow \infty$ for

  $$I_{2}(k)=\int_{0}^{\infty} e^{-\left(2 k^{2} / t+t^{2} / k\right)} d t$$

  [You may assume that $\Gamma(z)=\int_{0}^{\infty} t^{z-1} e^{-t} d t$ for $\operatorname{Re} z>0$,

  $$\text { and that } \left.\int_{-\infty}^{\infty} e^{-q t^{2}} d t=\sqrt{\pi / q} \text { for } q>0 .\right]$$

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• Paper 4, Section I, A

  #Classical Dynamics

  2014

  Consider a heavy symmetric top of mass $M$ with principal moments of inertia $I_{1}$, $I_{2}$ and $I_{3}$, where $I_{1}=I_{2} \neq I_{3}$. The top is pinned at point $P$, which is at a distance $l$ from the centre of mass, $C$, as shown in the figure.

  

  Its angular velocity in a body frame $\left(\mathbf{e}_{\mathbf{1}}, \mathbf{e}_{\mathbf{2}}, \mathbf{e}_{\mathbf{3}}\right)$ is given by

  $$\boldsymbol{\omega}=[\dot{\phi} \sin \theta \sin \psi+\dot{\theta} \cos \psi] \mathbf{e}_{1}+[\dot{\phi} \sin \theta \cos \psi-\dot{\theta} \sin \psi] \mathbf{e}_{2}+[\dot{\psi}+\dot{\phi} \cos \theta] \mathbf{e}_{3}$$

  where $\phi, \theta$ and $\psi$ are the Euler angles.

  (a) Assuming that $\left\{\mathbf{e}_{a}\right\}, a=1,2,3$, are chosen to be the principal axes, write down the Lagrangian of the top in terms of $\omega_{a}$ and the principal moments of inertia. Hence find the Lagrangian in terms of the Euler angles.

  (b) Find all conserved quantities. Show that $\omega_{3}$, the spin of the top, is constant.

  (c) By eliminating $\dot{\phi}$ and $\dot{\psi}$, derive a second-order differential equation for $\theta$.

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• Paper 3, Section I, A

  #Classical Dynamics

  2014

  (a) The action for a one-dimensional dynamical system with a generalized coordinate $q$ and Lagrangian $L$ is given by

  $$S=\int_{t_{1}}^{t_{2}} L(q, \dot{q}, t) d t$$

  State the principle of least action. Write the expression for the Hamiltonian in terms of the generalized velocity $\dot{q}$, the generalized momentum $p$ and the Lagrangian $L$. Use it to derive Hamilton's equations from the principle of least action.

  (b) The motion of a particle of charge $q$ and mass $m$ in an electromagnetic field with scalar potential $\phi(\mathbf{r}, t)$ and vector potential $\mathbf{A}(\mathbf{r}, t)$ is characterized by the Lagrangian

  $$L=\frac{m \dot{\mathbf{r}}^{2}}{2}-q(\phi-\dot{\mathbf{r}} \cdot \mathbf{A})$$

  (i) Write down the Hamiltonian of the particle.

  (ii) Consider a particle which moves in three dimensions in a magnetic field with $\mathbf{A}=(0, B x, 0)$, where $B$ is a constant. There is no electric field. Obtain Hamilton's equations for the particle.

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• Paper 2, Section I, A

  #Classical Dynamics

  2014

  The components of the angular velocity $\omega$ of a rigid body and of the position vector $\mathbf{r}$ are given in a body frame.

  (a) The kinetic energy of the rigid body is defined as

  $$T=\frac{1}{2} \int d^{3} \mathbf{r} \rho(\mathbf{r}) \dot{\mathbf{r}} \cdot \dot{\mathbf{r}}$$

  Given that the centre of mass is at rest, show that $T$ can be written in the form

  $$T=\frac{1}{2} I_{a b} \omega_{a} \omega_{b},$$

  where the explicit form of the tensor $I_{a b}$ should be determined.

  (b) Explain what is meant by the principal moments of inertia.

  (c) Consider a rigid body with principal moments of inertia $I_{1}, I_{2}$ and $I_{3}$, which are all unequal. Derive Euler's equations of torque-free motion

  $$\begin{aligned} &I_{1} \dot{\omega}_{1}=\left(I_{2}-I_{3}\right) \omega_{2} \omega_{3} \\ &I_{2} \dot{\omega}_{2}=\left(I_{3}-I_{1}\right) \omega_{3} \omega_{1} \\ &I_{3} \dot{\omega}_{3}=\left(I_{1}-I_{2}\right) \omega_{1} \omega_{2} \end{aligned}$$

  (d) The body rotates about the principal axis with moment of inertia $I_{1}$. Derive the condition for stable rotation.

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• Paper 1, Section I, A

  #Classical Dynamics

  2014

  Consider a one-dimensional dynamical system with generalized coordinate and momentum $(q, p)$.

  (a) Define the Poisson bracket $\{f, g\}$ of two functions $f(q, p, t)$ and $g(q, p, t)$.

  (b) Verify the Leibniz rule

  $$\{f g, h\}=f\{g, h\}+g\{f, h\}$$

  (c) Explain what is meant by a canonical transformation $(q, p) \rightarrow(Q, P)$.

  (d) State the condition for a transformation $(q, p) \rightarrow(Q, P)$ to be canonical in terms of the Poisson bracket $\{Q, P\}$. Use this to determine whether or not the following transformations are canonical:

  (i) $Q=\frac{q^{2}}{2}, P=\frac{p}{q}$,

  (ii) $Q=\tan q, P=p \cos q$,

  (iii) $Q=\sqrt{2 q} e^{t} \cos p, P=\sqrt{2 q} e^{-t} \sin p$.

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• Paper 4, Section II, A

  #Classical Dynamics

  2014

  (a) Consider a system with one degree of freedom, which undergoes periodic motion in the potential $V(q)$. The system's Hamiltonian is

  $$H(p, q)=\frac{p^{2}}{2 m}+V(q)$$

  (i) Explain what is meant by the angle and action variables, $\theta$ and $I$, of the system and write down the integral expression for the action variable $I$. Is $I$ conserved? Is $\theta$ conserved?

  (ii) Consider $V(q)=\lambda q^{6}$, where $\lambda$ is a positive constant. Find $I$ in terms of $\lambda$, the total energy $E$, the mass $M$, and a dimensionless constant factor (which you need not compute explicitly).

  (iii) Hence describe how $E$ changes with $\lambda$ if $\lambda$ varies slowly with time. Justify your answer.

  (b) Consider now a particle which moves in a plane subject to a central force-field $\mathbf{F}=-k r^{-2} \hat{\mathbf{r}}$.

  (i) Working in plane polar coordinates $(r, \phi)$, write down the Hamiltonian of the system. Hence deduce two conserved quantities. Prove that the system is integrable and state the number of action variables.

  (ii) For a particle which moves on an elliptic orbit find the action variables associated with radial and tangential motions. Can the relationship between the frequencies of the two motions be deduced from this result? Justify your answer.

  (iii) Describe how $E$ changes with $m$ and $k$ if one or both of them vary slowly with time.

  [You may use

  $$\int_{r_{1}}^{r_{2}}\left\{\left(1-\frac{r_{1}}{r}\right)\left(\frac{r_{2}}{r}-1\right)\right\}^{\frac{1}{2}} d r=\frac{\pi}{2}\left(r_{1}+r_{2}\right)-\pi \sqrt{r_{1} r_{2}}$$

  where $0<r_{1}<r_{2}$.]

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• Paper 2, Section II, A

  #Classical Dynamics

  2014

  A planar pendulum consists of a mass $m$ at the end of a light rod of length $l$. The pivot of the pendulum is attached to a bead of mass $M$, which slides along a horizontal rod without friction. The bead is connected to the ends of the horizontal rod by two identical springs of force constant $k$. The pivot constrains the pendulum to swing in the vertical plane through the horizontal rod. The horizontal rod is mounted on a bracket, so the system could rotate about the vertical axis which goes through its centre as shown in the figure.

  

  (a) Initially, the system is not allowed to rotate about the vertical axis.

  (i) Identify suitable generalized coordinates and write down the Lagrangian of the system.

  (ii) Write down expression(s) for any conserved quantities. Justify your answer.

  (iii) Derive the equations of motion.

  (iv) For $M=m / 2$ and $g m / k l=3$, find the frequencies of small oscillations around the stable equilibrium and the corresponding normal modes. Describe the respective motions of the system.

  (b) Assume now that the system is free to rotate about the vertical axis without friction. Write down the Lagrangian of the system. Identify and calculate the additional conserved quantity.

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• Paper 4, Section I, I

  #Coding and Cryptography

  2014

  Explain what is meant by a Bose-Ray Chaudhuri-Hocquenghem (BCH) code with design distance $\delta$. Prove that, for such a code, the minimum distance between code words is at least $\delta$. How many errors will the code detect? How many errors will it correct?

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• Paper 3, Section I, I

  #Coding and Cryptography

  2014

  Let $A$ be a random variable that takes values in the finite alphabet $\mathcal{A}$. Prove that there is a decodable binary code $c: \mathcal{A} \rightarrow\{0,1\}^{*}$ that satisfies

  $$H(A) \leqslant \mathbb{E}(l(A)) \leqslant H(A)+1$$

  where $l(a)$ is the length of the code word $c(a)$ and $H(A)$ is the entropy of $A$.

  Is it always possible to find such a code with $\mathbb{E}(l(A))=H(A) ?$ Justify your answer.

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• Paper 2, Section $I$, I

  #Coding and Cryptography

  2014

  Let $c: \mathcal{A} \rightarrow\{0,1\}^{*}$ be a decodable binary code defined on a finite alphabet $\mathcal{A}$. Let $l(a)$ be the length of the code word $c(a)$. Prove that

  $$\sum_{a \in \mathcal{A}} 2^{-l(a)} \leqslant 1$$

  Show that, for the decodable code $c: \mathcal{A} \rightarrow\{0,1\}^{*}$ described above, there is a prefixfree code $p: \mathcal{A} \rightarrow\{0,1\}^{*}$ with each code word $p(a)$ having length $l(a)$. [You may use, without proof, any standard results from the course.]

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• Paper 1, Section I, $4 \mathrm{I}$

  #Coding and Cryptography

  2014

  State and prove Gibbs' inequality.

  Show that, for a pair of discrete random variables $X$ and $Y$, each taking finitely many values, the joint entropy $H(X, Y)$ satisfies

  $$H(X, Y) \leqslant H(X)+H(Y)$$

  with equality precisely when $X$ and $Y$ are independent.

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• Paper 2, Section II, I

  #Coding and Cryptography

  2014

  What is the information capacity of a memoryless, time-independent channel? Compute the information capacity of a binary symmetric channel with probability $p$ of error. Show the steps in your computation.

  Binary digits are transmitted through a noisy channel, which is memoryless and time-independent. With probability $\alpha(0<\alpha<1)$ the digit is corrupted and noise is received, otherwise the digit is transmitted unchanged. So, if we denote the input by 0 and 1 and the output as $0, *$ and 1 with $*$ denoting the noise, the transition matrix is

  $$\left(\begin{array}{cc} 1-\alpha & 0 \\ \alpha & \alpha \\ 0 & 1-\alpha \end{array}\right)$$

  Compute the information capacity of this channel.

  Explain how to code a message for transmission through the channel described above, and how to decode it, so that the probability of error for each bit is arbitrarily small.

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• Paper 1, Section II, I

  #Coding and Cryptography

  2014

  Describe, briefly, either the RSA or the Elgamal public key cipher. You should explain, without proof, why it is believed to be difficult to break the cipher you describe.

  How can such a cipher be used to sign messages? You should explain how the intended recipient of the message can (a) know from whom it came; (b) know that the message has not been changed; and (c) demonstrate that the sender must have signed it.

  Let $I_{0}, I_{1}, \ldots, I_{N}$ be friendly individuals each of whom has a public key cipher. $I_{0}$ wishes to send a message to $I_{N}$ by passing it first to $I_{1}$, then $I_{1}$ passes it to $I_{2}, I_{2}$ to $I_{3}$, until finally it is received by $I_{N}$. At each stage the message can be modified to show from whom it was received and to whom it is sent. Devise a way in which these modifications can be made so that $I_{N}$ can be confident both of the content of the original message and that the message has been passed through the intermediaries $I_{1}, I_{2}, \ldots, I_{N-1}$ in that order and has not been modified by an enemy agent. Assume that it takes a negligible time to transmit a message from $I_{k}$ to $I_{k+1}$ for each $k$, but the time needed to modify a message is not negligible.

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• Paper 4, Section I, E

  #Cosmology

  2014

  A homogeneous and isotropic universe, with cosmological constant $\Lambda$, has expansion scale factor $a(t)$ and Hubble expansion rate $H=\dot{a} / a$. The universe contains matter with density $\rho$ and pressure $P$ which satisfy the positive-energy condition $\rho+3 P / c^{2} \geqslant 0$. The acceleration equation is

  $$\frac{\ddot{a}}{a}=-\frac{4 \pi G}{3}\left(\rho+3 P / c^{2}\right)+\frac{1}{3} \Lambda c^{2}$$

  If $\Lambda \leqslant 0$, show that

  $$\frac{d}{d t}\left(H^{-1}\right) \geqslant 1$$

  Deduce that $H \rightarrow \infty$ and $a \rightarrow 0$ at a finite time in the past or the future. What property of $H$ distinguishes the two cases?

  Give a simple counterexample with $\rho=P=0$ to show that this deduction fails to hold when $\Lambda>0$.

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• Paper 3, Section I, E

  #Cosmology

  2014

  Consider a finite sphere of zero-pressure material of uniform density $\rho(t)$ which expands with radius $r(t)=a(t) r_{0}$, where $r_{0}$ is an arbitary constant, due to the evolution of the expansion scale factor $a(t)$. The sphere has constant total mass $M$ and its radius satisfies

  $$\ddot{r}=-\frac{d \Phi}{d r}$$

  where

  $$\Phi(r)=-\frac{G M}{r}-\frac{1}{6} \Lambda r^{2} c^{2},$$

  with $\Lambda$ constant. Show that the scale factor obeys the equation

  $$\frac{\dot{a}^{2}}{a^{2}}=\frac{8 \pi G \rho}{3}-\frac{K c^{2}}{a^{2}}+\frac{1}{3} \Lambda c^{2},$$

  where $K$ is a constant. Explain why the sign, but not the magnitude, of $K$ is important. Find exact solutions of this equation for $a(t)$ when

  (i) $K=\Lambda=0$ and $\rho(t) \neq 0$,

  (ii) $\rho=K=0$ and $\Lambda>0$,

  (iii) $\rho=\Lambda=0$ and $K \neq 0$.

  Which two of the solutions (i)-(iii) are relevant for describing the evolution of the universe after the radiation-dominated era?

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• Paper 2, Section I, E

  #Cosmology

  2014

  A self-gravitating fluid with density $\rho$, pressure $P(\rho)$ and velocity $\mathbf{v}$ in a gravitational potential $\Phi$ obeys the equations

  $$\begin{aligned} \frac{\partial \rho}{\partial t}+\nabla \cdot(\rho \mathbf{v}) &=0 \\ \frac{\partial \mathbf{v}}{\partial t}+(\mathbf{v} \cdot \nabla) \mathbf{v}+\frac{\nabla P}{\rho}+\nabla \Phi &=\mathbf{0} \\ \nabla^{2} \Phi &=4 \pi G \rho \end{aligned}$$

  Assume that there exists a static constant solution of these equations with $\mathbf{v}=\mathbf{0}, \rho=\rho_{0}$ and $\Phi=\Phi_{0}$, for which $\nabla \Phi_{0}$ can be neglected. This solution is perturbed. Show that, to first order in the perturbed quantities, the density perturbations satisfy

  $$\frac{\partial^{2} \rho_{1}}{\partial t^{2}}=c_{s}^{2} \nabla^{2} \rho_{1}+4 \pi G \rho_{0} \rho_{1}$$

  where $\rho=\rho_{0}+\rho_{1}(\mathbf{x}, t)$ and $c_{s}^{2}=d P / d \rho$. Show that there are solutions to this equation of the form

  $$\rho_{1}(\mathbf{x}, t)=A \exp [-i \mathbf{k} \cdot \mathbf{x}+i \omega t]$$

  where $A, \omega$ and $\mathbf{k}$ are constants and

  $$\omega^{2}=c_{s}^{2} \mathbf{k} \cdot \mathbf{k}-4 \pi G \rho_{0}$$

  Interpret these solutions physically in the limits of small and large $|\mathbf{k}|$, explaining what happens to density perturbations on large and small scales, and determine the critical wavenumber that divides the two distinct behaviours of the perturbation.

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• Paper 1, Section I, E

  #Cosmology

  2014

  Which particle states are expected to be relativistic and which interacting when the temperature $T$ of the early universe satisfies (i) $10^{10} \mathrm{~K}<T<5 \times 10^{10} \mathrm{~K}$, (ii) $5 \times 10^{9} \mathrm{~K}<T<10^{10} \mathrm{~K}$, (iii) $T<5 \times 10^{9} \mathrm{~K}$ ?

  Calculate the total spin weight factor, $g_{*}$, of the relativistic particles and the total spin weight factor, $g_{I}$, of the interacting particles, in each of the three temperature intervals.

  What happens when the temperature falls below $5 \times 10^{9} \mathrm{~K} ?$ Calculate the ratio of the temperatures of neutrinos and photons. Find the effective value of $g_{*}$ after the universe cools below this temperature. [Note that the equilibrium entropy density is given by $s=\left(\rho c^{2}+P\right) / T$, where $\rho$ is the density and $P$ is the pressure.]

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• Paper 3, Section II, E

  #Cosmology

  2014

  The luminosity distance to an astronomical light source is given by $d_{L}=\chi / a(t)$, where $a(t)$ is the expansion scale factor and $\chi$ is the comoving distance in the universe defined by $d t=a(t) d \chi$. A zero-curvature Friedmann universe containing pressure-free matter and a cosmological constant with density parameters $\Omega_{m}$ and $\Omega_{\Lambda} \equiv 1-\Omega_{m}$, respectively, obeys the Friedmann equation

  $$H^{2}=H_{0}^{2}\left(\frac{\Omega_{m 0}}{a^{3}}+\Omega_{\Lambda 0}\right)$$

  where $H=(d a / d t) / a$ is the Hubble expansion rate of the universe and the subscript 0 denotes present-day values, with $a_{0} \equiv 1$.

  If $z$ is the redshift, show that

  $$d_{L}(z)=\frac{1+z}{H_{0}} \int_{0}^{z} \frac{d z^{\prime}}{\left[\left(1-\Omega_{\Lambda 0}\right)\left(1+z^{\prime}\right)^{3}+\Omega_{\Lambda 0}\right]^{1 / 2}}$$

  Find $d_{L}(z)$ when $\Omega_{\Lambda 0}=0$ and when $\Omega_{m 0}=0$. Roughly sketch the form of $d_{L}(z)$ for these two cases. What is the effect of a cosmological constant $\Lambda$ on the luminosity distance at a fixed value of $z$ ? Briefly describe how the relation between luminosity distance and redshift has been used to establish the acceleration of the expansion of the universe.

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• Paper 1, Section II, E

  #Cosmology

  2014

  What are the cosmological flatness and horizon problems? Explain what form of time evolution of the cosmological expansion scale factor $a(t)$ must occur during a period of inflationary expansion in a Friedmann universe. How can inflation solve the horizon and flatness problems? [You may assume an equation of state where pressure $P$ is proportional to density $\rho$.]

  The universe has Hubble expansion rate $H=\dot{a} / a$ and contains only a scalar field $\phi$ with self-interaction potential $V(\phi)>0$. The density and pressure are given by

  $$\begin{aligned} \rho &=\frac{1}{2} \dot{\phi}^{2}+V(\phi) \\ P &=\frac{1}{2} \dot{\phi}^{2}-V(\phi) \end{aligned}$$

  in units where $c=\hbar=1$. Show that the conservation equation

  $$\dot{\rho}+3 H(\rho+P)=0$$

  requires

  $$\ddot{\phi}+3 H \dot{\phi}+d V / d \phi=0 .$$

  If the Friedmann equation has the form

  $$3 H^{2}=8 \pi G \rho$$

  and the scalar-field potential has the form

  $$V(\phi)=V_{0} e^{-\lambda \phi}$$

  where $V_{0}$ and $\lambda$ are positive constants, show that there is an exact cosmological solution with

  $$\begin{aligned} &a(t) \propto t^{16 \pi G / \lambda^{2}} \\ &\phi(t)=\phi_{0}+\frac{2}{\lambda} \ln (t), \end{aligned}$$

  where $\phi_{0}$ is a constant. Find the algebraic relation between $\lambda, V_{0}$ and $\phi_{0}$. Show that a solution only exists when $0<\lambda^{2}<48 \pi G$. For what range of values of $\lambda^{2}$ does inflation occur? Comment on what happens when $\lambda \rightarrow 0$.

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• Paper 4, Section II, G

  #Differential Geometry

  2014

  Let $I=[0, l]$ be a closed interval, $k(s), \tau(s)$ smooth real valued functions on $I$ with $k$ strictly positive at all points, and $\mathbf{t}_{0}, \mathbf{n}_{0}, \mathbf{b}_{0}$ a positively oriented orthonormal triad of vectors in $\mathbf{R}^{3}$. An application of the fundamental theorem on the existence of solutions to ODEs implies that there exists a unique smooth family of triples of vectors $\mathbf{t}(s), \mathbf{n}(s), \mathbf{b}(s)$ for $s \in I$ satisfying the differential equations

  $$\mathbf{t}^{\prime}=k \mathbf{n}, \quad \mathbf{n}^{\prime}=-k \mathbf{t}-\tau \mathbf{b}, \quad \mathbf{b}^{\prime}=\tau \mathbf{n}$$

  with initial conditions $\mathbf{t}(0)=\mathbf{t}_{0}, \mathbf{n}(0)=\mathbf{n}_{0}$ and $\mathbf{b}(0)=\mathbf{b}_{0}$, and that $\{\mathbf{t}(s), \mathbf{n}(s), \mathbf{b}(s)\}$ forms a positively oriented orthonormal triad for all $s \in I$. Assuming this fact, consider $\alpha: I \rightarrow \mathbf{R}^{3}$ defined by $\alpha(s)=\int_{0}^{s} \mathbf{t}(t) d t$; show that $\alpha$ defines a smooth immersed curve parametrized by arc-length, which has curvature and torsion given by $k(s)$ and $\tau(s)$, and that $\alpha$ is uniquely determined by this property up to rigid motions of $\mathbf{R}^{3}$. Prove that $\alpha$ is a plane curve if and only if $\tau$ is identically zero.

  If $a>0$, calculate the curvature and torsion of the smooth curve given by

  $$\alpha(s)=(a \cos (s / c), a \sin (s / c), b s / c), \quad \text { where } c=\sqrt{a^{2}+b^{2}}$$

  Suppose now that $\alpha:[0,2 \pi] \rightarrow \mathbf{R}^{3}$ is a smooth simple closed curve parametrized by arc-length with curvature everywhere positive. If both $k$ and $\tau$ are constant, show that $k=1$ and $\tau=0$. If $k$ is constant and $\tau$ is not identically zero, show that $k>1$. Explain what it means for $\alpha$ to be knotted; if $\alpha$ is knotted and $\tau$ is constant, show that $k(s)>2$ for some $s \in[0,2 \pi]$. [You may use standard results from the course if you state them precisely.]

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• Paper 3, Section II, G

  #Differential Geometry

  2014

  Let $\alpha: I \rightarrow S$ be a parametrized curve on a smooth embedded surface $S \subset \mathbf{R}^{3}$. Define what is meant by a vector field $V$ along $\alpha$ and the concept of such a vector field being parallel. If $V$ and $W$ are both parallel vector fields along $\alpha$, show that the inner product $\langle V(t), W(t)\rangle$ is constant.

  Given a local parametrization $\phi: U \rightarrow S$, define the Christoffel symbols $\Gamma_{j k}^{i}$ on $U$. Given a vector $v_{0} \in T_{\alpha(0)} S$, prove that there exists a unique parallel vector field $V(t)$ along $\alpha$ with $V(0)=v_{0}$ (recall that $V(t)$ is called the parallel transport of $v_{0}$ along $\alpha$ ).

  Suppose now that the image of $\alpha$ also lies on another smooth embedded surface $S^{\prime} \subset \mathbf{R}^{3}$ and that $T_{\alpha(t)} S=T_{\alpha(t)} S^{\prime}$ for all $t \in I$. Show that parallel transport of a vector $v_{0}$ is the same whether calculated on $S$ or $S^{\prime}$. Suppose $S$ is the unit sphere in $\mathbf{R}^{3}$ with centre at the origin and let $\alpha:[0,2 \pi] \rightarrow S$ be the curve on $S$ given by

  $$\alpha(t)=(\sin \phi \cos t, \sin \phi \sin t, \cos \phi)$$

  for some fixed angle $\phi$. Suppose $v_{0} \in T_{P} S$ is the unit tangent vector to $\alpha$ at $P=\alpha(0)=$ $\alpha(2 \pi)$ and let $v_{1}$ be its image in $T_{P} S$ under parallel transport along $\alpha$. Show that the angle between $v_{0}$ and $v_{1}$ is $2 \pi \cos \phi$.

  [Hint: You may find it useful to consider the circular cone $S^{\prime}$ which touches the sphere $S$ along the curve $\alpha$.]

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• Paper 2, Section II, G

  #Differential Geometry

  2014

  Define the terms Gaussian curvature $K$ and mean curvature $H$ for a smooth embedded oriented surface $S \subset \mathbf{R}^{3}$. [You may assume the fact that the derivative of the Gauss map is self-adjoint.] If $K=H^{2}$ at all points of $S$, show that both $H$ and $K$ are locally constant. [Hint: Use the symmetry of second partial derivatives of the field of unit normal vectors.]

  If $K=H^{2}=0$ at all points of $S$, show that the unit normal vector $\mathbf{N}$ to $S$ is locally constant and that $S$ is locally contained in a plane. If $K=H^{2}$ is a strictly positive constant on $S$ and $\phi: U \rightarrow S$ is a local parametrization (where $U$ is connected) on $S$ with unit normal vector $\mathbf{N}(u, v)$ for $(u, v) \in U$, show that $\phi(u, v)+\mathbf{N}(u, v) / H$ is constant on $U$. Deduce that $S$ is locally contained in a sphere of radius $1 /|H|$.

  If $S$ is connected with $K=H^{2}$ at all points of $S$, deduce that $S$ is contained in either a plane or a sphere.

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• Paper 1, Section II, G

  #Differential Geometry

  2014

  Define the concepts of (smooth) manifold and manifold with boundary for subsets of $\mathbf{R}^{N}$.

  Let $X \subset \mathbf{R}^{6}$ be the subset defined by the equations

  $$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2}=1, \quad x_{4}^{2}-x_{5}^{2}-x_{6}^{2}=-1 .$$

  Prove that $X$ is a manifold of dimension four.

  For $a>0$, let $B(a) \subset \mathbf{R}^{6}$ denote the spherical ball $x_{1}^{2}+\ldots+x_{6}^{2} \leqslant a$. Prove that $X \cap B(a)$ is empty if $a<2$, is a manifold diffeomorphic to $S^{2} \times S^{1}$ if $a=2$, and is a manifold with boundary if $a>2$, with each component of the boundary diffeomorphic to $S^{2} \times S^{1}$.

  [You may quote without proof any general results from lectures that you may need.]

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• Paper 4, Section I, D

  #Dynamical Systems

  2014

  Consider the map $x_{n+1}=\lambda x_{n}\left(1-x_{n}^{2}\right)$ for $-1 \leqslant x_{n} \leqslant 1$. What is the maximum value, $\lambda_{\max }$, for which the interval $[-1,1]$ is mapped into itself?

  Analyse the first two bifurcations that occur as $\lambda$ increases from 0 towards $\lambda_{\max }$, including an identification of the values of $\lambda$ at which the bifurcation occurs and the type of bifurcation.

  What type of bifurcation do you expect as the third bifurcation? Briefly give your reasoning.

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• Paper 3, Section $I$, D

  #Dynamical Systems

  2014

  Define the Poincaré index of a closed curve $\mathcal{C}$ for a vector field $\mathbf{f}(\mathbf{x}), \mathbf{x} \in \mathbb{R}^{2}$.

  Explain carefully why the index of $\mathcal{C}$ is fully determined by the fixed points of the dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ that lie within $\mathcal{C}$.

  What is the Poincaré index for a closed curve $\mathcal{C}$ if it (a) encloses only a saddle point, (b) encloses only a focus and (c) encloses only a node?

  What is the Poincaré index for a closed curve $\mathcal{C}$ that is a periodic trajectory of the dynamical system?

  A dynamical system in $\mathbb{R}^{2}$ has 2 saddle points, 1 focus and 1 node. What is the maximum number of different periodic orbits? [For the purposes of this question, two orbits are said to be different if they enclose different sets of fixed points.]

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• Paper 2, Section I, D

  #Dynamical Systems

  2014

  Consider the system

  $$\begin{aligned} \dot{x} &=-x+y+y^{2} \\ \dot{y} &=\mu-x y \end{aligned}$$

  Show that when $\mu=0$ the fixed point at the origin has a stationary bifurcation.

  Find the centre subspace of the extended system linearised about $(x, y, \mu)=(0,0,0)$.

  Find an approximation to the centre manifold giving $y$ as a function of $x$ and $\mu$, including terms up to quadratic order.

  Hence deduce an expression for $\dot{x}$ on the centre manifold, and identify the type of bifurcation at $\mu=0$.

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• Paper 1, Section I, D

  #Dynamical Systems

  2014

  Consider the system

  $$\begin{aligned} &\dot{x}=y+x y \\ &\dot{y}=x-\frac{3}{2} y+x^{2} \end{aligned}$$

  Show that the origin is a hyperbolic fixed point and find the stable and unstable invariant subspaces of the linearised system.

  Calculate the stable and unstable manifolds correct to quadratic order, expressing $y$ as a function of $x$ for each.

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• Paper 4, Section II, D

  #Dynamical Systems

  2014

  A dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ has a fixed point at the origin. Define the terms Lyapunov stability, asymptotic stability and Lyapunov function with respect to this fixed point. State and prove Lyapunov's first theorem and state (without proof) La Salle's invariance principle.

  (a) Consider the system

  $$\begin{aligned} &\dot{x}=y \\ &\dot{y}=-y-x^{3}+x^{5} \end{aligned}$$

  Construct a Lyapunov function of the form $V=f(x)+g(y)$. Deduce that the origin is asymptotically stable, explaining your reasoning carefully. Find the greatest value of $y_{0}$ such that use of this Lyapunov function guarantees that the trajectory through $\left(0, y_{0}\right)$ approaches the origin as $t \rightarrow \infty$.

  (b) Consider the system

  $$\begin{aligned} &\dot{x}=x+4 y+x^{2}+2 y^{2}, \\ &\dot{y}=-3 x-3 y . \end{aligned}$$

  Show that the origin is asymptotically stable and that the basin of attraction of the origin includes the region $x^{2}+x y+y^{2}<\frac{1}{4}$.

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• Paper 3, Section II, D

  #Dynamical Systems

  2014

  Let $f: I \rightarrow I$ be a continuous one-dimensional map of an interval $I \subset \mathbb{R}$. Explain what is meant by saying that $f$ has a horseshoe.

  A map $g$ on the interval $[a, b]$ is a tent map if

  (i) $g(a)=a$ and $g(b)=a$;

  (ii) for some $c$ with $a<c<b, g$ is linear and increasing on the interval $[a, c]$, linear and decreasing on the interval $[c, b]$, and continuous at $c$.

  Consider the tent map defined on the interval $[0,1]$ by

  $$f(x)= \begin{cases}\mu x & 0 \leqslant x \leqslant \frac{1}{2} \\ \mu(1-x) & \frac{1}{2} \leqslant x \leqslant 1\end{cases}$$

  with $1<\mu \leqslant 2$. Find the corresponding expressions for $f^{2}(x)=f(f(x))$.

  Find the non-zero fixed point $x_{0}$ and the points $x_{-1}<\frac{1}{2}<x_{-2}$ that satisfy

  $$f^{2}\left(x_{-2}\right)=f\left(x_{-1}\right)=x_{0}=f\left(x_{0}\right) .$$

  Sketch graphs of $f$ and $f^{2}$ showing the points corresponding to $x_{-2}, x_{-1}$ and $x_{0}$. Indicate the values of $f$ and $f^{2}$ at their maxima and minima and also the gradients of each piece of their graphs.

  Identify a subinterval of $[0,1]$ on which $f^{2}$ is a tent map. Hence demonstrate that $f^{2}$ has a horseshoe if $\mu \geqslant 2^{1 / 2}$.

  Explain briefly why $f^{4}$ has a horseshoe when $\mu \geqslant 2^{1 / 4}$.

  Why are there periodic points of $f$ arbitrarily close to $x_{0}$ for $\mu \geqslant 2^{1 / 2}$, but no such points for $2^{1 / 4} \leqslant \mu<2^{1 / 2}$ ? Explain carefully any results or terms that you use.

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• Paper 4, Section II, 35C

  #Electrodynamics

  2014

  (i) The action $S$ for a point particle of rest mass $m$ and charge $q$ moving along a trajectory $x^{\mu}(\lambda)$ in the presence of an electromagnetic 4 -vector potential $A^{\mu}$ is

  $$S=-m c \int\left(-\eta_{\mu \nu} \frac{d x^{\mu}}{d \lambda} \frac{d x^{\nu}}{d \lambda}\right)^{1 / 2} d \lambda+q \int A_{\mu} \frac{d x^{\mu}}{d \lambda} d \lambda$$

  where $\lambda$ is an arbitrary parametrization of the path and $\eta_{\mu \nu}$ is the Minkowski metric. By varying the action with respect to $x^{\mu}(\lambda)$, derive the equation of motion $m \ddot{x}^{\mu}=q F^{\mu} \dot{x}^{\nu}$, where $F_{\mu \nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu}$ and overdots denote differentiation with respect to proper time for the particle.

  (ii) The particle moves in constant electric and magnetic fields with non-zero Cartesian components $E_{z}=E$ and $B_{y}=B$, with $B>E / c>0$ in some inertial frame. Verify that a suitable 4-vector potential has components

  $$A^{\mu}=(0,0,0,-B x-E t)$$

  in that frame.

  Find the equations of motion for $x, y, z$ and $t$ in terms of proper time $\tau$. For the case of a particle that starts at rest at the spacetime origin at $\tau=0$, show that

  $$\ddot{z}+\frac{q^{2}}{m^{2}}\left(B^{2}-\frac{E^{2}}{c^{2}}\right) z=\frac{q E}{m} .$$

  Find the trajectory $x^{\mu}(\tau)$ and sketch its projection onto the $(x, z)$ plane.

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• Paper 3, Section II, C

  #Electrodynamics

  2014

  The 4-vector potential $A^{\mu}(t, \mathbf{x})$ (in the Lorenz gauge $\partial_{\mu} A^{\mu}=0$ ) due to a localised source with conserved 4-vector current $J^{\mu}$ is

  $$A^{\mu}(t, \mathbf{x})=\frac{\mu_{0}}{4 \pi} \int \frac{J^{\mu}\left(t_{\mathrm{ret}}, \mathbf{x}^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|} d^{3} \mathbf{x}^{\prime}$$

  where $t_{\text {ret }}=t-\left|\mathbf{x}-\mathbf{x}^{\prime}\right| / c$. For a source that varies slowly in time, show that the spatial components of $A^{\mu}$ at a distance $r=|\mathbf{x}|$ that is large compared to the spatial extent of the source are

  $$\left.\mathbf{A}(t, \mathbf{x}) \approx \frac{\mu_{0}}{4 \pi r} \frac{d \mathbf{P}}{d t}\right|_{t-r / c}$$

  where $\mathbf{P}$ is the electric dipole moment of the source, which you should define. Explain what is meant by the far-field region, and calculate the leading-order part of the magnetic field there.

  A point charge $q$ moves non-relativistically in a circle of radius $a$ in the $(x, y)$ plane with angular frequency $\omega$ (such that $a \omega \ll c$ ). Show that the magnetic field in the far-field at the point $\mathbf{x}$ with spherical polar coordinates $r, \theta$ and $\phi$ has components along the $\theta$ and $\phi$ directions given by

  $$\begin{aligned} &B_{\theta} \approx-\frac{\mu_{0} \omega^{2} q a}{4 \pi r c} \sin [\omega(t-r / c)-\phi] \\ &B_{\phi} \approx \frac{\mu_{0} \omega^{2} q a}{4 \pi r c} \cos [\omega(t-r / c)-\phi] \cos \theta \end{aligned}$$

  Calculate the total power radiated by the charge.

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• Paper 1, Section II, 36C

  #Electrodynamics

  2014

  (i) Starting from the field-strength tensor $F_{\mu \nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu}$, where $A^{\mu}=(\phi / c, \mathbf{A})$ is the 4-vector potential with components such that

  $$\mathbf{E}=-\frac{\partial \mathbf{A}}{\partial t}-\boldsymbol{\nabla} \phi \quad \text { and } \quad \mathbf{B}=\boldsymbol{\nabla} \times \mathbf{A}$$

  derive the transformation laws for the components of the electric field $\mathbf{E}$ and the magnetic field $\mathbf{B}$ under the standard Lorentz boost $x^{\prime \mu}=\Lambda_{\nu}^{\mu} x^{\nu}$ with

  $$\Lambda_{\nu}^{\mu}=\left(\begin{array}{cccc} \gamma & -\gamma \beta & 0 & 0 \\ -\gamma \beta & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)$$

  (ii) Two point charges, each with electric charge $q$, are at rest and separated by a distance $d$ in some inertial frame $S$. By transforming the fields from the rest frame $S$, calculate the magnitude and direction of the force between the two charges in an inertial frame in which the charges are moving with speed $\beta c$ in a direction perpendicular to their separation.

  (iii) The 4-force for a particle with 4-momentum $p^{\mu}$ is $F^{\mu}=d p^{\mu} / d \tau$, where $\tau$ is proper time. Show that the components of $F^{\mu}$ in an inertial frame in which the particle has 3 -velocity $\mathbf{v}$ are

  $$F^{\mu}=\gamma(\mathbf{F} \cdot \mathbf{v} / c, \mathbf{F})$$

  where $\gamma=\left(1-\mathbf{v} \cdot \mathbf{v} / c^{2}\right)^{-1 / 2}$ and $\mathbf{F}$ is the 3-force acting on the particle. Hence verify that your result in (ii) above is consistent with Lorentz transforming the electromagnetic 3 -force from the rest frame $S$.

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• Paper 4, Section II, B

  #Fluid Dynamics II

  2014

  An incompressible fluid of density $\rho$ and kinematic viscosity $\nu$ is confined in a channel with rigid stationary walls at $y=\pm h$. A spatially uniform pressure gradient $-G \cos \omega t$ is applied in the $x$-direction. What is the physical significance of the dimensionless number $S=\omega h^{2} / \nu ?$

  Assuming that the flow is unidirectional and time-harmonic, obtain expressions for the velocity profile and the total flux. [You may leave your answers as the real parts of complex functions.]

  In each of the limits $S \rightarrow 0$ and $S \rightarrow \infty$, find and sketch the flow profiles, find leading-order asymptotic expressions for the total flux, and give a physical interpretation.

  Suppose now that $G=0$ and that the channel walls oscillate in their own plane with velocity $U \cos \omega t$ in the $x$-direction. Without explicit calculation of the solution, sketch the flow profile in each of the limits $S \rightarrow 0$ and $S \rightarrow \infty$.

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• Paper 2, Section II, B

  #Fluid Dynamics II

  2014

  Air is blown over the surface of a large, deep reservoir of water in such a way as to exert a tangential stress in the $x$-direction of magnitude $K x^{2}$ for $x>0$, with $K>0$. The water is otherwise at rest and occupies the region $y>0$. The surface $y=0$ remains flat.

  Find order-of-magnitude estimates for the boundary-layer thickness $\delta(x)$ and tangential surface velocity $U(x)$ in terms of the relevant physical parameters.

  Using the boundary-layer equations, find the ordinary differential equation governing the dimensionless function $f$ defined in the streamfunction

  $$\psi(x, y)=U(x) \delta(x) f(\eta), \quad \text { where } \eta=y / \delta(x)$$

  What are the boundary conditions on $f$ ?

  Does $f \rightarrow 0$ as $\eta \rightarrow \infty$ ? Why, or why not?

  The total horizontal momentum flux $P(X)$ across the vertical line $x=X$ is proportional to $X^{a}$ for $X>0$. Find the exponent $a$. By considering the steadiness of the momentum balance in the region $0<x<X$, explain why the value of $a$ is consistent with the form of the stress exerted on the boundary.

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• Paper 3, Section II, B

  #Fluid Dynamics II

  2014

  A rigid sphere of radius $a$ falls under gravity through an incompressible fluid of density $\rho$ and viscosity $\mu$ towards a rigid horizontal plane. The minimum gap $h_{0}(t)$ between the sphere and the plane satisfies $h_{0} \ll a$. Find an approximation for the gap thickness $h(r, t)$ between the sphere and the plane in the region $r \ll a$, where $r$ is the distance from the axis of symmetry.

  For a prescribed value of $\dot{h}_{0}=d h_{0} / d t$, use lubrication theory to find the radial velocity and the fluid pressure in the region $r \ll a$. Explain why the approximations of lubrication theory require $h_{0} \ll a$ and $\rho h_{0} \dot{h}_{0} \ll \mu$.

  Calculate the total vertical force due to the motion that is exerted by the fluid on the sphere. Deduce that if the sphere is settling under its own weight (corrected for buoyancy) then $h_{0}(t)$ decreases exponentially. What is the exponential decay rate for a solid sphere of density $\rho_{s}$ in a fluid of density $\rho_{f}$ ?

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• Paper 1, Section II, B

  #Fluid Dynamics II

  2014

  A particle of arbitrary shape and volume $4 \pi a^{3} / 3$ moves at velocity $\mathbf{U}(t)$ through an unbounded incompressible fluid of density $\rho$ and viscosity $\mu$. The Reynolds number of the flow is very small so that the inertia of the fluid can be neglected. Show that the particle experiences a force $\mathbf{F}(t)$ due to the surface stresses given by

  $$F_{i}(t)=-\mu a A_{i j} U_{j}(t)$$

  where $A_{i j}$ is a dimensionless second-rank tensor determined solely by the shape and orientation of the particle. State the reason why $A_{i j}$ must be positive definite.

  Show further that, if the particle has the same reflectional symmetries as a cube, then

  $$A_{i j}=\lambda \delta_{i j}$$

  Let $b$ be the radius of the smallest sphere that contains the particle (still assuming cubic symmetry). By considering the Stokes flow associated with this sphere, suitably extended, and using the minimum dissipation theorem (which should be stated carefully), show that

  $$\lambda \leqslant 6 \pi b / a .$$

  [You may assume the expression for the Stokes drag on a sphere.]

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• Paper 4, Section I, B

  #Further Complex Methods

  2014

  Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a function such that

  $$f\left(z+\omega_{1}\right)=f(z), \quad f\left(z+\omega_{2}\right)=f(z)$$

  where $\omega_{1}, \omega_{2} \in \mathbb{C} \backslash\{0\}$ and $\omega_{1} / \omega_{2}$ is not real. Show that if $f$ is analytic on $\mathbb{C}$ then it is a constant. [Liouville's theorem may be used if stated.] Give an example of a non-constant meromorphic function which satisfies (1).

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• Paper 3, Section I, B

  #Further Complex Methods

  2014

  State the conditions for a point $z=z_{0}$ to be a regular singular point of a linear second-order homogeneous ordinary differential equation in the complex plane.

  Find all singular points of the Airy equation

  $$w^{\prime \prime}(z)-z w(z)=0$$

  and determine whether they are regular or irregular.

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• Paper 1, Section $I$, B

  #Further Complex Methods

  2014

  Show that the Cauchy-Riemann equations for $f: \mathbb{C} \rightarrow \mathbb{C}$ are equivalent to

  $$\frac{\partial f}{\partial \bar{z}}=0 \text {, }$$

  where $z=x+i y$, and $\partial / \partial \bar{z}$ should be defined in terms of $\partial / \partial x$ and $\partial / \partial y$. Use Green's theorem, together with the formula $d z d \bar{z}=-2 i d x d y$, to establish the generalised Cauchy formula

  $$\oint_{\gamma} f(z, \bar{z}) d z=-\iint_{D} \frac{\partial f}{\partial \bar{z}} d z d \bar{z}$$

  where $\gamma$ is a contour in the complex plane enclosing the region $D$ and $f$ is sufficiently differentiable.

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• Paper 2, Section I, B

  #Further Complex Methods

  2014

  Suppose $z=0$ is a regular singular point of a linear second-order homogeneous ordinary differential equation in the complex plane. Define the monodromy matrix $M$ around $z=0$.

  Demonstrate that if

  $$M=\left(\begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\right)$$

  then the differential equation admits a solution of the form $a(z)+b(z) \log z$, where $a(z)$ and $b(z)$ are single-valued functions.

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• Paper 2, Section II, 14B

  #Further Complex Methods

  2014

  Use the Euler product formula

  $$\Gamma(z)=\lim _{n \rightarrow \infty} \frac{n ! n^{z}}{z(z+1) \ldots(z+n)}$$

  to show that:

  (i) $\Gamma(z+1)=z \Gamma(z)$;

  (ii) $\frac{1}{\Gamma(z)}=z e^{\gamma z} \prod_{k=1}^{\infty}\left(1+\frac{z}{k}\right) e^{-z / k}$, where $\gamma=\lim _{n \rightarrow \infty}\left(1+\frac{1}{2}+\cdots+\frac{1}{n}-\log n\right)$.

  Deduce that

  $$\frac{d}{d z} \log (\Gamma(z))=-\gamma-\frac{1}{z}+z \sum_{k=1}^{\infty} \frac{1}{k(z+k)}$$

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• Paper 1, Section II, 14B

  #Further Complex Methods

  2014

  Obtain solutions of the second-order ordinary differential equation

  $$z w^{\prime \prime}-w=0$$

  in the form

  $$w(z)=\int_{\gamma} f(t) e^{-z t} d t$$

  where the function $f$ and the choice of contour $\gamma$ should be determined from the differential equation.

  Show that a non-trivial solution can be obtained by choosing $\gamma$ to be a suitable closed contour, and find the resulting solution in this case, expressing your answer in the form of a power series.

  Describe a contour $\gamma$ that would provide a second linearly independent solution for the case $\operatorname{Re}(z)>0$.

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• Paper 4, Section II, H

  #Galois Theory

  2014

  (i) Let $G$ be a finite subgroup of the multiplicative group of a field. Show that $G$ is cyclic.

  (ii) Let $\Phi_{n}(X)$ be the $n$th cyclotomic polynomial. Let $p$ be a prime not dividing $n$, and let $L$ be a splitting field for $\Phi_{n}$ over $\mathbb{F}_{p}$. Show that $L$ has $p^{m}$ elements, where $m$ is the least positive integer such that $p^{m} \equiv 1(\bmod n)$.

  (iii) Find the degrees of the irreducible factors of $X^{35}-1$ over $\mathbb{F}_{2}$, and the number of factors of each degree.

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• Paper 3, Section II, H

  #Galois Theory

  2014

  Let $L / K$ be an algebraic extension of fields, and $x \in L$. What does it mean to say that $x$ is separable over $K$ ? What does it mean to say that $L / K$ is separable?

  Let $K=\mathbb{F}_{p}(t)$ be the field of rational functions over $\mathbb{F}_{p}$.

  (i) Show that if $x$ is inseparable over $K$ then $K(x)$ contains a $p$ th root of $t$.

  (ii) Show that if $L / K$ is finite there exists $n \geqslant 0$ and $y \in L$ such that $y^{p^{n}}=t$ and $L / K(y)$ is separable.

  Show that $Y^{2}+t Y+t$ is an irreducible separable polynomial over the field of rational functions $K=\mathbb{F}_{2}(t)$. Find the degree of the splitting field of $X^{4}+t X^{2}+t$ over $K$.

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• Paper 2, Section II, H

  #Galois Theory

  2014

  Describe the Galois correspondence for a finite Galois extension $L / K$.

  Let $L$ be the splitting field of $X^{4}-2$ over $\mathbb{Q}$. Compute the Galois group $G$ of $L / \mathbb{Q}$. For each subgroup of $G$, determine the corresponding subfield of $L$.

  Let $L / K$ be a finite Galois extension whose Galois group is isomorphic to $S_{n}$. Show that $L$ is the splitting field of a separable polynomial of degree $n$.

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• Paper 1, Section II, 18H

  #Galois Theory

  2014

  What is meant by the statement that $L$ is a splitting field for $f \in K[X] ?$

  Show that if $f \in K[X]$, then there exists a splitting field for $f$ over $K$. Explain the sense in which a splitting field for $f$ over $K$ is unique.

  Determine the degree $[L: K]$ of a splitting field $L$ of the polynomial $f=X^{4}-4 X^{2}+2$ over $K$ in the cases (i) $K=\mathbb{Q}$, (ii) $K=\mathbb{F}_{5}$, and (iii) $K=\mathbb{F}_{7}$.

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• Paper 4, Section II, E

  #General Relativity

  2014

  A plane-wave spacetime has line element

  $$d s^{2}=H d u^{2}+2 d u d v+d x^{2}+d y^{2}$$

  where $H=x^{2}-y^{2}$. Show that the line element is unchanged by the coordinate transformation

  $$u=\bar{u}, \quad v=\bar{v}+\bar{x} e^{\bar{u}}-\frac{1}{2} e^{2 \bar{u}}, \quad x=\bar{x}-e^{\bar{u}}, \quad y=\bar{y}$$

  Show more generally that the line element is unchanged by coordinate transformations of the form

  $$u=\bar{u}+a, \quad v=\bar{v}+b \bar{x}+c, \quad x=\bar{x}+p, \quad y=\bar{y}$$

  where $a, b, c$ and $p$ are functions of $\bar{u}$, which you should determine and which depend in total on four parameters (arbitrary constants of integration).

  Deduce (without further calculation) that the line element is unchanged by a 6parameter family of coordinate transformations, of which a 5 -parameter family leave invariant the surfaces $u=$ constant.

  For a general coordinate transformation $x^{a}=x^{a}\left(\bar{x}^{b}\right)$, give an expression for the transformed Ricci tensor $\bar{R}_{c d}$ in terms of the Ricci tensor $R_{a b}$ and the transformation matrices $\frac{\partial x^{a}}{\partial \bar{x}^{c}}$. Calculate $\bar{R}_{\bar{x} \bar{x}}$ when the transformation is given by $(*)$ and deduce that $R_{v v}=R_{v x}$

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• Paper 2, Section II, E

  #General Relativity

  2014

  Show how the geodesic equations and hence the Christoffel symbols $\Gamma^{a} b c$ can be obtained from a Lagrangian.

  In units with $c=1$, the FLRW spacetime line element is

  $$d s^{2}=-d t^{2}+a^{2}(t)\left(d x^{2}+d y^{2}+d z^{2}\right)$$

  Show that $\Gamma_{01}^{1}=\dot{a} / a$.

  You are given that, for the above metric, $G_{0}{ }^{0}=-3 \dot{a}^{2} / a^{2}$ and $G_{1}^{1}=-2 \ddot{a} / a-\dot{a}^{2} / a^{2}$, where $G_{a}^{b}$ is the Einstein tensor, which is diagonal. Verify by direct calculation that $\nabla_{b} G_{a}^{b}=0$.

  Solve the vacuum Einstein equations in the presence of a cosmological constant to determine the form of $a(t)$.

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• Paper 3, Section II, $37 \mathrm{E}$

  #General Relativity

  2014

  The vector field $V^{a}$ is the normalised $\left(V_{a} V^{a}=-c^{2}\right)$ tangent to a congruence of timelike geodesics, and $B_{a b}=\nabla_{b} V_{a}$.

  Show that:

  (i) $V^{a} B_{a b}=V^{b} B_{a b}=0$;

  (ii) $V^{c} \nabla_{c} B_{a b}=-B_{b}^{c} B_{a c}-R_{a c b}^{d} V^{c} V_{d}$.

  [You may use the Ricci identity $\nabla_{c} \nabla_{b} X_{a}=\nabla_{b} \nabla_{c} X_{a}-R_{a c b}^{d} X_{d}$.]

  Now assume that $B_{a b}$ is symmetric and let $\theta=B_{a}{ }^{a}$. By writing $B_{a b}=\widetilde{B}_{a b}+\frac{1}{4} \theta g_{a b}$, or otherwise, show that

  $$\frac{d \theta}{d \tau} \leqslant-\frac{1}{4} \theta^{2}-R_{00}$$

  where $R_{00}=R_{a b} V^{a} V^{b}$ and $\frac{d \theta}{d \tau} \equiv V^{a} \nabla_{a} \theta$. [You may use without proof the result that \left.\widetilde{B}_{a b} \widetilde{B}^{a b} \geqslant 0 .\right]

  Assume, in addition, that the stress-energy tensor $T_{a b}$ takes the perfect-fluid form $\left(\rho+p / c^{2}\right) V_{a} V_{b}+p g_{a b}$ and that $\rho c^{2}+3 p>0$. Show that

  $$\frac{d \theta^{-1}}{d \tau}>\frac{1}{4}$$

  and deduce that, if $\theta(0)<0$, then $|\theta(\tau)|$ will become unbounded for some value of $\tau$ less than $4 /|\theta(0)|$.

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• Paper 1, Section II, 37E

  #General Relativity

  2014

  For a timelike geodesic in the equatorial plane $\left(\theta=\frac{1}{2} \pi\right)$ of the Schwarzschild spacetime with line element

  $$d s^{2}=-\left(1-r_{s} / r\right) c^{2} d t^{2}+\left(1-r_{s} / r\right)^{-1} d r^{2}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)$$

  derive the equation

  $$\frac{1}{2} \dot{r}^{2}+V(r)=\frac{1}{2}(E / c)^{2}$$

  where

  $$\frac{2 V(r)}{c^{2}}=1-\frac{r_{s}}{r}+\frac{h^{2}}{c^{2} r^{2}}-\frac{h^{2} r_{s}}{c^{2} r^{3}}$$

  and $h$ and $E$ are constants. The dot denotes the derivative with respect to an affine parameter $\tau$ satisfying $c^{2} d \tau^{2}=-d s^{2}$.

  Given that there is a stable circular orbit at $r=R$, show that

  $$\frac{h^{2}}{c^{2}}=\frac{R^{2} \epsilon}{2-3 \epsilon}$$

  where $\epsilon=r_{s} / R$.

  Compute $\Omega$, the orbital angular frequency (with respect to $\tau$ ).

  Show that the angular frequency $\omega$ of small radial perturbations is given by

  $$\frac{\omega^{2} R^{2}}{c^{2}}=\frac{\epsilon(1-3 \epsilon)}{2-3 \epsilon}$$

  Deduce that the rate of precession of the perihelion of the Earth's orbit, $\Omega-\omega$, is approximately $3 \Omega^{3} T^{2}$, where $T$ is the time taken for light to travel from the Sun to the Earth. [You should assume that the Earth's orbit is approximately circular, with $r_{s} / R \ll 1$ and $\left.E \simeq c^{2} .\right]$

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• Paper 4, Section I, F

  #Geometry and Groups

  2014

  Define the limit set $\Lambda(G)$ of a Kleinian group $G$. Assuming that $G$ has no finite orbit in $\mathbb{H}^{3} \cup S_{\infty}^{2}$, and that $\Lambda(G) \neq \emptyset$, prove that if $E \subset \mathbb{C} \cup\{\infty\}$ is any non-empty closed set which is invariant under $G$, then $\Lambda(G) \subset E$.

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• Paper 3, Section I, F

  #Geometry and Groups

  2014

  Let $\mathbb{H}^{2}$ denote the hyperbolic plane, and $T \subset \mathbb{H}^{2}$ be a non-degenerate triangle, i.e. the bounded region enclosed by three finite-length geodesic arcs. Prove that the three angle bisectors of $T$ meet at a point.

  Must the three vertices of $T$ lie on a hyperbolic circle? Justify your answer.

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• Paper 2, Section I, F

  #Geometry and Groups

  2014

  Let $g, h$ be non-identity Möbius transformations. Prove that $g$ and $h$ commute if and only if one of the following holds:

  1. $\operatorname{Fix}(g)=\operatorname{Fix}(h)$;

  2. $g, h$ are involutions each of which exchanges the other's fixed points.

  Give an example to show that the second case can occur.

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• Paper 1, Section I, F

  #Geometry and Groups

  2014

  Let $G \leqslant S O(3)$ be a finite group. Suppose $G$ does not preserve any plane in $\mathbb{R}^{3}$. Show that for any point $p$ in the unit sphere $S^{2} \subset \mathbb{R}^{3}$, the stabiliser $\operatorname{Stab}_{G}(p)$ contains at most 5 elements.

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• Paper 1, Section II, F

  #Geometry and Groups

  2014

  Prove that an orientation-preserving isometry of the ball-model of hyperbolic space $\mathbb{H}^{3}$ which fixes the origin is an element of $S O(3)$. Hence, or otherwise, prove that a finite subgroup of the group of orientation-preserving isometries of hyperbolic space $\mathbb{H}^{3}$ has a common fixed point.

  Can an infinite non-cyclic subgroup of the isometry group of $\mathbb{H}^{3}$ have a common fixed point? Can any such group be a Kleinian group? Justify your answers.

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• Paper 4, Section II, F

  #Geometry and Groups

  2014

  Define the $s$-dimensional Hausdorff measure $\mathcal{H}^{s}(F)$ of a set $F \subset \mathbb{R}^{N}$. Explain briefly how properties of this measure may be used to define the Hausdorff dimension $\operatorname{dim}_{H}(F)$ of such a set.

  Prove that the limit sets of conjugate Kleinian groups have equal Hausdorff dimension. Hence, or otherwise, prove that there is no subgroup of $\mathbb{P} S L(2, \mathbb{R})$ which is conjugate in $\mathbb{P} S L(2, \mathbb{C})$ to $\mathbb{P} S L(2, \mathbb{Z} \oplus \mathbb{Z} i)$.

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• Paper 4, Section II, I

  #Graph Theory

  2014

  Define the Ramsey number $R^{(r)}(s, t)$. What is the value of $R^{(1)}(s, t)$ ? Prove that $R^{(r)}(s, t) \leqslant 1+R^{(r-1)}\left(R^{(r)}(s-1, t), R^{(r)}(s, t-1)\right)$ holds for $r \geqslant 2$ and deduce that $R^{(r)}(s, t)$ exists.

  Show that $R^{(2)}(3,3)=6$ and that $R^{(2)}(3,4)=9$.

  Show that $7 \leqslant R^{(3)}(4,4) \leqslant 19$. [Hint: For the lower bound, choose a suitable subset $U$ and colour e red if $|U \cap e|$ is odd.]

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• Paper 3, Section II, I

  #Graph Theory

  2014

  Prove that $\chi(G) \leqslant \Delta(G)+1$ for every graph $G$. Prove further that, if $\kappa(G) \geqslant 3$, then $\chi(G) \leqslant \Delta(G)$ unless $G$ is complete.

  Let $k \geqslant 2$. A graph $G$ is said to be $k$-critical if $\chi(G)=k+1$, but $\chi(G-v)=k$ for every vertex $v$ of $G$. Show that, if $G$ is $k$-critical, then $\kappa(G) \geqslant 2$.

  Let $k \geqslant 2$, and let $H$ be the graph $K_{k+1}$ with an edge removed. Show that $H$ has the following property: it has two vertices which receive the same colour in every $k$-colouring of $H$. By considering two copies of $H$, construct a $k$-colourable graph $G$ of order $2 k+1$ with the following property: it has three vertices which receive the same colour in every $k$-colouring of $G$.

  Construct, for all integers $k \geqslant 2$ and $\ell \geqslant 2$, a $k$-critical graph $G$ of order $\ell k+1$ with $\kappa(G)=2 .$

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• Paper 2, Section II, I

  #Graph Theory

  2014

  Let $k$ and $n$ be integers with $1 \leqslant k<n$. Show that every connected graph of order $n$, in which $d(u)+d(v) \geqslant k$ for every pair $u, v$ of non-adjacent vertices, contains a path of length $k$.

  Let $k$ and $n$ be integers with $1 \leqslant k \leqslant n$. Show that a graph of order $n$ that contains no path of length $k$ has at most $(k-1) n / 2$ edges, and that this value is achieved only if $k$ divides $n$ and $G$ is the union of $n / k$ disjoint copies of $K_{k}$. [Hint: Proceed by induction on $n$ and consider a vertex of minimum degree.]

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• Paper 1, Section II, I

  #Graph Theory

  2014

  Show that a graph is bipartite if and only if all of its cycles are of even length.

  Show that a bridgeless plane graph is bipartite if and only if all of its faces are of even length.

  Let $G$ be an Eulerian plane graph. Show that the faces of $G$ can be coloured with two colours so that no two contiguous faces have the same colour. Deduce that it is possible to assign a direction to each edge of $G$ in such a way that the edges around each face form a directed cycle.

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• Paper 3, Section II, D

  #Integrable Systems

  2014

  What does it mean to say that a finite-dimensional Hamiltonian system is integrable? State the Arnold-Liouville theorem.

  A six-dimensional dynamical system with coordinates $\left(x_{1}, x_{2}, x_{3}, y_{1}, y_{2}, y_{3}\right)$ is governed by the differential equations

  $$\frac{\mathrm{d} x_{i}}{\mathrm{~d} t}=-\frac{1}{2 \pi} \sum_{j \neq i} \frac{\Gamma_{j}\left(y_{i}-y_{j}\right)}{\left(x_{i}-x_{j}\right)^{2}+\left(y_{i}-y_{j}\right)^{2}}, \quad \frac{\mathrm{d} y_{i}}{\mathrm{~d} t}=\frac{1}{2 \pi} \sum_{j \neq i} \frac{\Gamma_{j}\left(x_{i}-x_{j}\right)}{\left(x_{i}-x_{j}\right)^{2}+\left(y_{i}-y_{j}\right)^{2}}$$

  for $i=1,2,3$, where $\left\{\Gamma_{i}\right\}_{i=1}^{3}$ are positive constants. Show that these equations can be written in the form

  $$\Gamma_{i} \frac{\mathrm{d} x_{i}}{\mathrm{~d} t}=\frac{\partial F}{\partial y_{i}}, \quad \Gamma_{i} \frac{\mathrm{d} y_{i}}{\mathrm{~d} t}=-\frac{\partial F}{\partial x_{i}}, \quad i=1,2,3$$

  for an appropriate function $F$. By introducing the coordinates

  $$\mathbf{q}=\left(x_{1}, x_{2}, x_{3}\right), \quad \mathbf{p}=\left(\Gamma_{1} y_{1}, \Gamma_{2} y_{2}, \Gamma_{3} y_{3}\right),$$

  show that the system can be written in Hamiltonian form

  $$\frac{\mathrm{d} \mathbf{q}}{\mathrm{d} t}=\frac{\partial H}{\partial \mathbf{p}}, \quad \frac{\mathrm{d} \mathbf{p}}{\mathrm{d} t}=-\frac{\partial H}{\partial \mathbf{q}}$$

  for some Hamiltonian $H=H(\mathbf{q}, \mathbf{p})$ which you should determine.

  Show that the three functions

  $$A=\sum_{i=1}^{3} \Gamma_{i} x_{i}, \quad B=\sum_{i=1}^{3} \Gamma_{i} y_{i}, \quad C=\sum_{i=1}^{3} \Gamma_{i}\left(x_{i}^{2}+y_{i}^{2}\right)$$

  are first integrals of the Hamiltonian system.

  Making use of the fundamental Poisson brackets $\left\{q_{i}, q_{j}\right\}=\left\{p_{i}, p_{j}\right\}=0$ and $\left\{q_{i}, p_{j}\right\}=\delta_{i j}$, show that

  $$\{A, C\}=2 B, \quad\{B, C\}=-2 A$$

  Hence show that the Hamiltonian system is integrable.

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• Paper 2, Section II, D

  #Integrable Systems

  2014

  Let $u=u(x)$ be a smooth function that decays rapidly as $|x| \rightarrow \infty$ and let $L=-\partial_{x}^{2}+u(x)$ denote the associated Schrödinger operator. Explain very briefly each of the terms appearing in the scattering data

  $$S=\left\{\left\{\chi_{n}, c_{n}\right\}_{n=1}^{N}, R(k)\right\},$$

  associated with the operator $L$. What does it mean to say $u(x)$ is reflectionless?

  Given $S$, define the function

  $$F(x)=\sum_{n=1}^{N} c_{n}^{2} e^{-\chi_{n} x}+\frac{1}{2 \pi} \int_{-\infty}^{\infty} e^{\mathrm{i} k x} R(k) \mathrm{d} k$$

  If $K=K(x, y)$ is the unique solution to the GLM equation

  $$K(x, y)+F(x+y)+\int_{x}^{\infty} K(x, z) F(z+y) \mathrm{d} z=0$$

  what is the relationship between $u(x)$ and $K(x, x)$ ?

  Now suppose that $u=u(x, t)$ is time dependent and that it solves the KdV equation $u_{t}+u_{x x x}-6 u u_{x}=0$. Show that $L=-\partial_{x}^{2}+u(x, t)$ obeys the Lax equation

  $$L_{t}=[L, A], \quad \text { where } A=4 \partial_{x}^{3}-3\left(u \partial_{x}+\partial_{x} u\right) .$$

  Show that the discrete eigenvalues of $L$ are time independent.

  In what follows you may assume the time-dependent scattering data take the form

  $$S(t)=\left\{\left\{\chi_{n}, c_{n} e^{4 \chi_{n}^{3} t}\right\}_{n=1}^{N}, R(k, 0) e^{8 \mathrm{i} k^{3} t}\right\} .$$

  Show that if $u(x, 0)$ is reflectionless, then the solution to the KdV equation takes the form

  $$u(x, t)=-2 \frac{\partial^{2}}{\partial x^{2}} \log [\operatorname{det} A(x, t)]$$

  where $A$ is an $N \times N$ matrix which you should determine.

  Assume further that $R(k, 0)=k^{2} f(k)$, where $f$ is smooth and decays rapidly at infinity. Show that, for any fixed $x$,

  $$\int_{-\infty}^{\infty} e^{\mathrm{i} k x} R(k, 0) e^{8 \mathrm{i} k^{3} t} \mathrm{~d} k=O\left(t^{-1}\right) \quad \text { as } t \rightarrow \infty$$

  Comment briefly on the significance of this result.

  [You may assume $\frac{1}{\operatorname{det} A} \frac{\mathrm{d}}{\mathrm{d} x}(\operatorname{det} A)=\operatorname{tr}\left(A^{-1} \frac{\mathrm{d} A}{\mathrm{~d} x}\right)$ for a non-singular matrix $A(x)$.]

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• Paper 1, Section II, D

  #Integrable Systems

  2014

  Consider the coordinate transformation

  $$g^{\epsilon}:(x, u) \mapsto(\tilde{x}, \tilde{u})=(x \cos \epsilon-u \sin \epsilon, x \sin \epsilon+u \cos \epsilon)$$

  Show that $g^{\epsilon}$ defines a one-parameter group of transformations. Define what is meant by the generator $V$ of a one-parameter group of transformations and compute it for the above case.

  Now suppose $u=u(x)$. Explain what is meant by the first prolongation $\mathrm{pr}^{(1)} g^{\epsilon}$ of $g^{\epsilon}$. Compute $\mathrm{pr}^{(1)} g^{\epsilon}$ in this case and deduce that

  $$\mathrm{pr}^{(1)} V=V+\left(1+u_{x}^{2}\right) \frac{\partial}{\partial u_{x}}$$

  Similarly find $\mathrm{pr}^{(2)} V$.

  Define what is meant by a Lie point symmetry of the first-order differential equation $\Delta\left[x, u, u_{x}\right]=0$. Describe this condition in terms of the vector field that generates the Lie point symmetry. Consider the case

  $$\Delta\left[x, u, u_{x}\right] \equiv u_{x}-\frac{u+x f\left(x^{2}+u^{2}\right)}{x-u f\left(x^{2}+u^{2}\right)}$$

  where $f$ is an arbitrary smooth function of one variable. Using $(\star)$, show that $g^{\epsilon}$ generates a Lie point symmetry of the corresponding differential equation.

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• Paper 3, Section II, G

  #Linear Analysis

  2014

  (i) State carefully the theorems of Stone-Weierstrass and Arzelá-Ascoli (work with real scalars only).

  (ii) Let $\mathcal{F}$ denote the family of functions on $[0,1]$ of the form

  $$f(x)=\sum_{n=1}^{\infty} a_{n} \sin (n x)$$

  where the $a_{n}$ are real and $\left|a_{n}\right| \leqslant 1 / n^{3}$ for all $n \in \mathbb{N}$. Prove that any sequence in $\mathcal{F}$ has a subsequence that converges uniformly on $[0,1]$.

  (iii) Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function such that $f(0)=0$ and $f^{\prime}(0)$ exists. Show that for each $\varepsilon>0$ there exists a real polynomial $p$ having only odd powers, i.e. of the form

  $$p(x)=a_{1} x+a_{3} x^{3}+\cdots+a_{2 m-1} x^{2 m-1},$$

  such that $\sup _{x \in[0,1]}|f(x)-p(x)|<\varepsilon$. Show that the same holds without the assumption that $f$ is differentiable at 0 .

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• Paper 1, Section II, G

  #Linear Analysis

  2014

  Let $X$ and $Y$ be normed spaces. What is an isomorphism between $X$ and $Y$ ? Show that a bounded linear map $T: X \rightarrow Y$ is an isomorphism if and only if $T$ is surjective and there is a constant $c>0$ such that $\|T x\| \geqslant c\|x\|$ for all $x \in X$. Show that if there is an isomorphism $T: X \rightarrow Y$ and $X$ is complete, then $Y$ is complete.

  Show that two normed spaces of the same finite dimension are isomorphic. [You may assume without proof that any two norms on a finite-dimensional space are equivalent.] Briefly explain why this implies that every finite-dimensional space is complete, and every closed and bounded subset of a finite-dimensional space is compact.

  Let $Z$ and $F$ be subspaces of a normed space $X$ with $Z \cap F=\{0\}$. Assume that $Z$ is closed in $X$ and $F$ is finite-dimensional. Prove that $Z+F$ is closed in $X$. [Hint: First show that the function $x \mapsto d(x, Z)=\inf \{\|x-z\|: z \in Z\}$ restricted to the unit sphere of F achieves its minimum.]

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• Paper 2, Section II, G

  #Linear Analysis

  2014

  (a) Let $X$ and $Y$ be Banach spaces, and let $T: X \rightarrow Y$ be a surjective linear map. Assume that there is a constant $c>0$ such that $\|T x\| \geqslant c\|x\|$ for all $x \in X$. Show that $T$ is continuous. [You may use any standard result from general Banach space theory provided you clearly state it.] Give an example to show that the assumption that $X$ and $Y$ are complete is necessary.

  (b) Let $C$ be a closed subset of a Banach space $X$ such that

  (i) $x_{1}+x_{2} \in C$ for each $x_{1}, x_{2} \in C$;

  (ii) $\lambda x \in C$ for each $x \in C$ and $\lambda>0$;

  (iii) for each $x \in X$, there exist $x_{1}, x_{2} \in C$ such that $x=x_{1}-x_{2}$.

  Prove that, for some $M>0$, the unit ball of $X$ is contained in the closure of the set

  $$\left\{x_{1}-x_{2}: x_{i} \in C, \quad\left\|x_{i}\right\| \leqslant M(i=1,2)\right\} .$$

  [You may use without proof any version of the Baire Category Theorem.] Deduce that, for some $K>0$, every $x \in X$ can be written as $x=x_{1}-x_{2}$ with $x_{i} \in C$ and $\left\|x_{i}\right\| \leqslant K\|x\|(i=1,2) .$

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• Paper 4, Section II, G

  #Linear Analysis

  2014

  Define the spectrum $\sigma(T)$ and the approximate point spectrum $\sigma_{\mathrm{ap}}(T)$ of a bounded linear operator $T$ on a Banach space. Prove that $\sigma_{\mathrm{ap}}(T) \subset \sigma(T)$ and that $\sigma(T)$ is a closed and bounded subset of $\mathbb{C}$. [You may assume without proof that the set of invertible operators is open.]

  Let $T$ be a hermitian operator on a non-zero Hilbert space. Prove that $\sigma(T)$ is not empty

  Let $K$ be a non-empty, compact subset of $\mathbb{C}$. Show that there is a bounded linear operator $T: \ell_{2} \rightarrow \ell_{2}$ with $\sigma(T)=K .$ [You may assume without proof that a compact metric space is separable.]

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• Paper 4, Section II, I

  #Logic and Set Theory

  2014

  Explain what is meant by a chain-complete poset. State the Bourbaki-Witt fixedpoint theorem.

  We call a poset $(P, \leqslant)$ Bourbakian if every order-preserving map $f: P \rightarrow P$ has a least fixed point $\mu(f)$. Suppose $P$ is Bourbakian, and let $f, g: P \rightrightarrows P$ be order-preserving maps with $f(x) \leqslant g(x)$ for all $x \in P$; show that $\mu(f) \leqslant \mu(g)$. [Hint: Consider the function $h: P \rightarrow P$ defined by $h(x)=f(x)$ if $x \leqslant \mu(g), h(x)=\mu(g)$ otherwise.]

  Suppose $P$ is Bourbakian and $f: \alpha \rightarrow P$ is an order-preserving map from an ordinal to $P$. Show that there is an order-preserving map $g: P \rightarrow P$ whose fixed points are exactly the upper bounds of the set $\{f(\beta) \mid \beta<\alpha\}$, and deduce that this set has a least upper bound.

  Let $C$ be a chain with no greatest member. Using the Axiom of Choice and Hartogs' Lemma, show that there is an order-preserving map $f: \alpha \rightarrow C$, for some ordinal $\alpha$, whose image has no upper bound in $C$. Deduce that any Bourbakian poset is chain-complete.

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• Paper 3, Section II, I

  #Logic and Set Theory

  2014

  Explain what is meant by a structure for a first-order signature $\Sigma$, and describe briefly how first-order terms and formulae in the language over $\Sigma$ are interpreted in a structure. Suppose that $A$ and $B$ are $\Sigma$-structures, and that $\phi$ is a conjunction of atomic formulae over $\Sigma$ : show that an $n$-tuple $\left(\left(a_{1}, b_{1}\right), \ldots,\left(a_{n}, b_{n}\right)\right)$ belongs to the interpretation $\llbracket \phi \rrbracket_{A \times B}$ of $\phi$ in $A \times B$ if and only if $\left(a_{1}, \ldots, a_{n}\right) \in \llbracket \phi \rrbracket_{A}$ and $\left(b_{1}, \ldots, b_{n}\right) \in \llbracket \phi \rrbracket B$.

  A first-order theory $\mathbb{T}$ is called regular if its axioms all have the form

  $$(\forall \vec{x})(\phi \Rightarrow(\exists \vec{y}) \psi),$$

  where $\vec{x}$ and $\vec{y}$ are (possibly empty) strings of variables and $\phi$ and $\psi$ are conjunctions of atomic formulae (possibly the empty conjunction $T$ ). Show that if $A$ and $B$ are models of a regular theory $\mathbb{T}$, then so is $A \times B$.

  Now suppose that $\mathbb{T}$ is a regular theory, and that a sentence of the form

  $$(\forall \vec{x})\left(\phi \Rightarrow\left(\psi_{1} \vee \psi_{2} \vee \cdots \vee \psi_{n}\right)\right)$$

  is derivable from the axioms of $\mathbb{T}$, where $\phi$ and the $\psi_{i}$ are conjunctions of atomic formulae. Show that the sentence $(\forall \vec{x})\left(\phi \Rightarrow \psi_{i}\right)$ is derivable for some $i$. [Hint: Suppose not, and use the Completeness Theorem to obtain a suitable family of $\mathbb{T}$-models $A_{1}, \ldots, A_{n}$.]

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• Paper 2, Section II, I

  #Logic and Set Theory

  2014

  Write down the recursive definitions of ordinal addition, multiplication and exponentiation. Show that, for any nonzero ordinal $\alpha$, there exist unique ordinals $\beta, \gamma$ and $n$ such that $\alpha=\omega^{\beta} . n+\gamma, \gamma<\omega^{\beta}$ and $0<n<\omega$.

  Hence or otherwise show that $\alpha$ (that is, the set of ordinals less than $\alpha$ ) is closed under addition if and only if $\alpha=\omega^{\beta}$ for some $\beta$. Show also that an infinite ordinal $\alpha$ is closed under multiplication if and only if $\alpha=\omega^{\left(\omega^{\gamma}\right)}$ for some $\gamma$.

  [You may assume the standard laws of ordinal arithmetic, and the fact that $\alpha \leqslant \omega^{\alpha}$ for all $\alpha$.]

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• Paper 1, Section II, I

  #Logic and Set Theory

  2014

  Explain what is meant by saying that a binary relation $r \subseteq a \times a$ is well-founded. Show that $r$ is well-founded if and only if, for any set $b$ and any function $f: \mathcal{P} b \rightarrow b$, there exists a unique function $g: a \rightarrow b$ satisfying

  $$g(x)=f(\{g(y) \mid\langle y, x\rangle \in r\})$$

  for all $x \in a$. [Hint: For 'if', it suffices to take $b=\{0,1\}$, with $f: \mathcal{P} b \rightarrow b$ defined by $\left.f\left(b^{\prime}\right)=1 \Leftrightarrow 1 \in b^{\prime} .\right]$

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• Paper 4, Section I, B

  #Mathematical Biology

  2014

  The concentration $c(x, t)$ of a chemical in one dimension obeys the equations

  $$\frac{\partial c}{\partial t}=\frac{\partial}{\partial x}\left(c^{2} \frac{\partial c}{\partial x}\right), \quad \int_{-\infty}^{\infty} c(x, t) d x=1$$

  State the physical interpretation of each equation.

  Seek a similarity solution of the form $c=t^{\alpha} f(\xi)$, where $\xi=t^{\beta} x$. Find equations involving $\alpha$ and $\beta$ from the differential equation and the integral. Show that these are satisfied by $\alpha=\beta=-1 / 4$.

  Find the solution for $f(\xi)$. Find and sketch the solution for $c(x, t)$.

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• Paper 3, Section $I$, B

  #Mathematical Biology

  2014

  An epidemic model is given by

  $$\begin{aligned} &\frac{d S}{d t}=-r I S \\ &\frac{d I}{d t}=+r I S-a I \end{aligned}$$

  where $S(t)$ are the susceptibles, $I(t)$ are the infecteds, and $a$ and $r$ are positive parameters. The basic reproduction ratio is defined as $R_{0}=r N / a$, where $N$ is the total population size. Find a condition on $R_{0}$ for an epidemic to be possible if, initially, $S \approx N$ and $I$ is small but non-zero.

  Now suppose a proportion $p$ of the population was vaccinated (with a completely effective vaccine) so that initially $S \approx(1-p) N$. On a sketch of the $\left(R_{0}, p\right)$ plane, mark the regions where an epidemic is still possible, where the vaccination will prevent an epidemic, and where no vaccination was necessary.

  For the case when an epidemic is possible, show that $\sigma$, the proportion of the initially susceptible population that has not been infected by the end of an epidemic, satisfies

  $$\sigma-\frac{1}{(1-p) R_{0}} \log \sigma \approx 1$$

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• Paper 2, Section I, B

  #Mathematical Biology

  2014

  Consider an experiment where two or three individuals are added to a population with probability $\lambda_{2}$ and $\lambda_{3}$ respectively per unit time. The death rate in the population is a constant $\beta$ per individual per unit time.

  Write down the master equation for the probability $p_{n}(t)$ that there are $n$ individuals in the population at time $t$. From this, derive an equation for $\frac{\partial \phi}{\partial t}$, where $\phi$ is the generating function

  $$\phi(s, t)=\sum_{n=0}^{\infty} s^{n} p_{n}(t)$$

  Find the solution for $\phi$ in steady state, and show that the mean and variance of the population size are given by

  $$\langle n\rangle=3 \frac{\lambda_{3}}{\beta}+2 \frac{\lambda_{2}}{\beta}, \quad \operatorname{var}(n)=6 \frac{\lambda_{3}}{\beta}+3 \frac{\lambda_{2}}{\beta} .$$

  Hence show that, for a free choice of $\lambda_{2}$ and $\lambda_{3}$ subject to a given target mean, the experimenter can minimise the variance by only adding two individuals at a time.

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• Paper 1, Section I, B

  #Mathematical Biology

  2014

  A population model for two species is given by

  $$\begin{aligned} &\frac{d N}{d t}=a N-b N P-k N^{2} \\ &\frac{d P}{d t}=-d P+c N P \end{aligned}$$

  where $a, b, c, d$ and $k$ are positive parameters. Show that this may be rescaled to

  $$\begin{aligned} &\frac{d u}{d \tau}=u(1-v-\beta u) \\ &\frac{d v}{d \tau}=-\alpha v(1-u) \end{aligned}$$

  and give $\alpha$ and $\beta$ in terms of the original parameters.

  For $\beta<1$ find all fixed points in $u \geqslant 0, v \geqslant 0$, and analyse their stability. Assuming that both populations are present initially, what does this suggest will be the long-term outcome?

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• Paper 3, Section II, B

  #Mathematical Biology

  2014

  A discrete-time model for breathing is given by

  $$\begin{aligned} V_{n+1} &=\alpha C_{n-k}, \\ C_{n+1}-C_{n} &=\gamma-\beta V_{n+1}, \end{aligned}$$

  where $V_{n}$ is the volume of each breath in time step $n$ and $C_{n}$ is the concentration of carbon dioxide in the blood at the end of time step $n$. The parameters $\alpha, \beta$ and $\gamma$ are all positive. Briefly explain the biological meaning of each of the above equations.

  Find the steady state. For $k=0$ and $k=1$ determine the stability of the steady state.

  For general (integer) $k>1$, by seeking parameter values when the modulus of a perturbation to the steady state is constant, find the range of parameters where the solution is stable. What is the periodicity of the constant-modulus solution at the edge of this range? Comment on how the size of the range depends on $k$.

  This can be developed into a more realistic model by changing the term $-\beta V_{n+1}$ to $-\beta C_{n} V_{n+1}$ in (2). Briefly explain the biological meaning of this change. Show that for both $k=0$ and $k=1$ the new steady state is stable if $0<a<1$, where $a=\sqrt{\alpha \beta \gamma}$.

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• Paper 2, Section II, B

  #Mathematical Biology

  2014

  An activator-inhibitor system is described by the equations

  $$\begin{aligned} \frac{\partial u}{\partial t} &=\frac{a u}{v}-u^{2}+d_{1} \frac{\partial^{2} u}{\partial x^{2}} \\ \frac{\partial v}{\partial t} &=v^{2}-\frac{v}{u^{2}}+d_{2} \frac{\partial^{2} v}{\partial x^{2}} \end{aligned}$$

  where $a, d_{1}, d_{2}>0$.

  Find the range of $a$ for which the spatially homogeneous system has a stable equilibrium solution with $u>0$ and $v>0$. Determine when the equilibrium is a stable focus, and sketch the phase diagram for this case (restricting attention to $u>0$ and $v>0)$.

  For the case when the homogeneous system is stable, consider spatial perturbations proportional to $\cos (k x)$ of the solution found above. Briefly explain why the system will be stable to spatial perturbations with very small or very large $k$. Find conditions for the system to be unstable to a spatial perturbation (for some range of $k$ which need not be given). Sketch the region satisfying these conditions in the $\left(a, d_{1} / d_{2}\right)$ plane.

  Find $k_{c}$, the critical wavenumber at the onset of instability, in terms of $a$ and $d_{1}$.

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• Paper 4, Section II, F

  #Number Fields

  2014

  Explain what is meant by an integral basis for a number field. Splitting into the cases $d \equiv 1(\bmod 4)$ and $d \equiv 2,3(\bmod 4)$, find an integral basis for $K=\mathbb{Q}(\sqrt{d})$ where $d \neq 0,1$ is a square-free integer. Justify your answer.

  Find the fundamental unit in $\mathbb{Q}(\sqrt{13})$. Determine all integer solutions to the equation $x^{2}+x y-3 y^{2}=17$.

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• Paper 2, Section II, F

  #Number Fields

  2014

  (i) Show that each prime ideal in a number field $K$ divides a unique rational prime $p$. Define the ramification index and residue class degree of such an ideal. State and prove a formula relating these numbers, for all prime ideals dividing a given rational prime $p$, to the degree of $K$ over $\mathbb{Q}$.

  (ii) Show that if $\zeta_{n}$ is a primitive $n$th root of unity then $\prod_{j=1}^{n-1}\left(1-\zeta_{n}^{j}\right)=n$. Deduce that if $n=p q$, where $p$ and $q$ are distinct primes, then $1-\zeta_{n}$ is a unit in $\mathbb{Z}\left[\zeta_{n}\right]$.

  (iii) Show that if $K=\mathbb{Q}\left(\zeta_{p}\right)$ where $p$ is prime, then any prime ideal of $K$ dividing $p$ has ramification index at least $p-1$. Deduce that $[K: \mathbb{Q}]=p-1$.

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• Paper 1, Section II, F

  #Number Fields

  2014

  State a result involving the discriminant of a number field that implies that the class group is finite.

  Use Dedekind's theorem to factor $2,3,5$ and 7 into prime ideals in $K=\mathbb{Q}(\sqrt{-34})$. By factoring $1+\sqrt{-34}$ and $4+\sqrt{-34}$, or otherwise, prove that the class group of $K$ is cyclic, and determine its order.

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• Paper 4, Section I, F

  #Number Theory

  2014

  State the Chinese Remainder Theorem.

  Find all solutions to the simultaneous congruences

  $$\begin{aligned} &x \equiv 2 \quad(\bmod 3) \\ &x \equiv 3(\bmod 5) \\ &x \equiv 5(\bmod 7) \end{aligned}$$

  A positive integer is said to be square-free if it is the product of distinct primes. Show that there are 100 consecutive numbers that are not square-free.

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• Paper 3, Section I, $1 F$

  #Number Theory

  2014

  Show that the continued fraction for $\sqrt{51}$ is $[7 ; \overline{7,14}]$.

  Hence, or otherwise, find positive integers $x$ and $y$ that satisfy the equation $x^{2}-51 y^{2}=1 .$

  Are there integers $x$ and $y$ such that $x^{2}-51 y^{2}=-1$ ?

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• Paper 2, Section I, F

  #Number Theory

  2014

  Show that

  $$\sum_{p \leqslant x} \frac{1}{p} \geqslant \log \log x-\frac{1}{2}$$

  Deduce that there are infinitely many primes.

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• Paper 1, Section I, $1 F$

  #Number Theory

  2014

  Define what it means for a number $N$ to be a pseudoprime to the base $b$.

  Show that if there is a base $b$ to which $N$ is not a pseudoprime, then $N$ is a pseudoprime to at most half of all possible bases.

  Let $n$ be an integer greater than 1 such that $F_{n}=2^{2^{n}}+1$ is composite. Show that $F_{n}$ is a pseudoprime to the base 2 .

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• Paper 4, Section II, F

  #Number Theory

  2014

  Define the Legendre and Jacobi symbols.

  State the law of quadratic reciprocity for the Legendre symbol.

  State the law of quadratic reciprocity for the Jacobi symbol, and deduce it from the corresponding result for the Legendre symbol.

  Let $p$ be a prime with $p \equiv 1(\bmod 4)$. Prove that the sum of the quadratic residues in the set $\{1,2, \ldots, p-1\}$ is equal to the sum of the quadratic non-residues in this set.

  For which primes $p$ is 7 a quadratic residue?

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• Paper 3, Section II, F

  #Number Theory

  2014

  State and prove Lagrange's theorem about polynomial congruences modulo a prime.

  Define the Euler totient function $\phi$.

  Let $p$ be a prime and let $d$ be a positive divisor of $p-1$. Show that there are exactly $\phi(d)$ elements of $(\mathbb{Z} / p \mathbb{Z})^{\times}$with order $d .$

  Deduce that $(\mathbb{Z} / p \mathbb{Z})^{\times}$is cyclic.

  Let $g$ be a primitive root modulo $p^{2}$. Show that $g$ must be a primitive root modulo $p$.

  Let $g$ be a primitive root modulo $p$. Must it be a primitive root modulo $p^{2}$ ? Give a proof or a counterexample.

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• Paper 4, Section II, D

  #Numerical Analysis

  2014

  Let $A$ be a real symmetric $n \times n$ matrix with $n$ distinct real eigenvalues $\lambda_{1}<\lambda_{2}<$ $\cdots<\lambda_{n}$ and a corresponding orthogonal basis of normalized real eigenvectors $\left\{\mathbf{w}_{i}\right\}_{i=1}^{n}$.

  (i) Let $s \in \mathbb{R}$ satisfy $s<\lambda_{1}$. Given a unit vector $\mathbf{x}^{(0)} \in \mathbb{R}^{n}$, the iteration scheme

  $$\begin{gathered} (A-s I) \mathbf{y}=\mathbf{x}^{(k)} \\ \mathbf{x}^{(k+1)}=\mathbf{y} /\|\mathbf{y}\| \end{gathered}$$

  generates a sequence of vectors $\mathbf{x}^{(k+1)}$ for $k=0,1,2, \ldots$. Assuming that $\mathbf{x}^{(0)}=\sum c_{i} \mathbf{w}_{i}$ with $c_{1} \neq 0$, prove that $\mathbf{x}^{(k)}$ tends to $\pm \mathbf{w}_{1}$ as $k \rightarrow \infty$. What happens to $\mathbf{x}^{(k)}$ if $s>\lambda_{1}$ ? [Consider all cases.]

  (ii) Describe how to implement an inverse-iteration algorithm to compute the eigenvalues and eigenvectors of $A$, given some initial estimates for the eigenvalues.

  (iii) Let $n=2$. For iterates $\mathbf{x}^{(k)}$ of an inverse-iteration algorithm with a fixed value of $s \neq \lambda_{1}, \lambda_{2}$, show that if

  $$\mathbf{x}^{(k)}=\left(\mathbf{w}_{1}+\epsilon_{k} \mathbf{w}_{2}\right) /\left(1+\epsilon_{k}^{2}\right)^{1 / 2}$$

  where $\left|\epsilon_{k}\right|$ is small, then $\left|\epsilon_{k+1}\right|$ is of the same order of magnitude as $\left|\epsilon_{k}\right|$.

  (iv) Let $n=2$ still. Consider the iteration scheme

  $$s_{k}=\left(\mathbf{x}^{(k)}, A \mathbf{x}^{(k)}\right), \quad\left(A-s_{k} I\right) \mathbf{y}=\mathbf{x}^{(k)}, \quad \mathbf{x}^{(k+1)}=\mathbf{y} /\|\mathbf{y}\|$$

  for $k=0,1,2, \ldots$, where $(,$, denotes the inner product. Show that with this scheme $\left|\epsilon_{k+1}\right|=\left|\epsilon_{k}\right|^{3} .$

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• Paper 2, Section II, D

  #Numerical Analysis

  2014

  Consider the one-dimensional advection equation

  $$u_{t}=u_{x}, \quad-\infty<x<\infty, \quad t \geqslant 0$$

  subject to an initial condition $u(x, 0)=\varphi(x)$. Consider discretization of this equation with finite differences on an equidistant space-time $\left\{(m h, n k), m \in \mathbb{Z}, n \in \mathbb{Z}^{+}\right\}$with step size $h>0$ in space and step size $k>0$ in time. Define the Courant number $\mu$ and explain briefly how such a discretization can be used to derive numerical schemes in which solutions $u_{m}^{n} \approx u(m h, n k), m \in \mathbb{Z}$ and $n \in \mathbb{Z}^{+}$satisfy equations of the form

  $$\sum_{i=r}^{s} a_{i} u_{m+i}^{n+1}=\sum_{i=r}^{s} b_{i} u_{m+i}^{n}$$

  where the coefficients $a_{i}, b_{i}$ are independent of $m, n$.

  (i) Define the order of a numerical scheme such as (1). Define what a convergent numerical scheme is. Explain the notion of stability and state the Lax equivalence theorem that connects convergence and stability of numerical schemes for linear partial differential equations.

  (ii) Consider the following example of (1):

  $$u_{m}^{n+1}=u_{m}^{n}+\frac{\mu}{2}\left(u_{m+1}^{n}-u_{m-1}^{n}\right)+\frac{\mu^{2}}{2}\left(u_{m+1}^{n}-2 u_{m}^{n}+u_{m-1}^{n}\right) .$$

  Determine conditions on $\mu$ such that the scheme $(2)$ is stable and convergent. What is the order of this scheme?

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• Paper 3, Section II, D

  #Numerical Analysis

  2014

  Consider the linear system

  $$A x=b$$

  where $A \in \mathbb{R}^{n \times n}$ and $b, x \in \mathbb{R}^{n}$.

  (i) Define the Jacobi iteration method with relaxation parameter $\omega$ for solving (1).

  (ii) Assume that $A$ is a symmetric positive-definite matrix whose diagonal part $D$ is such that the matrix $2 D-A$ is also positive definite. Prove that the relaxed Jacobi iteration method always converges if the relaxation parameter $\omega$ is equal to $1 .$

  (iii) Let $A$ be the tridiagonal matrix with diagonal elements $a_{i i}=\alpha$ and off-diagonal elements $a_{i+1, i}=a_{i, i+1}=\beta$, where $0<\beta<\frac{1}{2} \alpha$. For which values of $\omega$ (expressed in terms of $\alpha$ and $\beta$ ) does the relaxed Jacobi iteration method converge? What choice of $\omega$ gives the optimal convergence speed?

  [You may quote without proof any relevant results about the convergence of iterative methods and about the eigenvalues of matrices.]

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• Paper 1, Section II, 40D

  #Numerical Analysis

  2014

  (i) Consider the numerical approximation of the boundary-value problem

  $$\begin{gathered} u^{\prime \prime}=f, \quad u:[0,1] \rightarrow \mathbb{R}, \\ u(0)=\varphi_{0}, \quad u(1)=\varphi_{1}, \end{gathered}$$

  where $\varphi_{0}, \varphi_{1}$ are given constants and $f$ is a given smooth function on $[0,1]$. A grid $\left\{x_{1}, x_{2}, \ldots, x_{N}\right\}, N \geqslant 3$, on $[0,1]$ is given by

  $$x_{1}=\alpha_{1} h, \quad x_{i}=x_{i-1}+h \text { for } i=2, \ldots, N-1, \quad x_{N}=1-\alpha_{2} h$$

  where $0<\alpha_{1}, \alpha_{2}<1, \alpha_{1}+\alpha_{2}=1$ and $h=1 / N$. Derive finite-difference approximations for $u^{\prime \prime}\left(x_{i}\right)$, for $i=1, \ldots, N$, using at most one neighbouring grid point of $x_{i}$ on each side. Hence write down a numerical scheme to solve the problem, displaying explicitly the entries of the system matrix $A$ in the resulting system of linear equations $A u=b$, $A \in \mathbb{R}^{N \times N}, u, b \in \mathbb{R}^{N}$. What is the overall order of this numerical scheme? Explain briefly one strategy by which the order could be improved with the same grid.

  (ii) Consider the numerical approximation of the boundary-value problem

  $$\begin{aligned} &\nabla^{2} u=f, \quad u: \Omega \rightarrow \mathbb{R}, \\ &u(x)=0 \text { for all } x \in \partial \Omega \end{aligned}$$

  where $\Omega \subset \mathbb{R}^{2}$ is an arbitrary, simply connected bounded domain with smooth boundary $\partial \Omega$, and $f$ is a given smooth function. Define the 9-point formula used to approximate the Laplacian. Using this formula and an equidistant grid inside $\Omega$, define a numerical scheme for which the system matrix is symmetric and negative definite. Prove that the system matrix of your scheme has these properties for all choices of ordering of the grid points.

  Part II, $2014 \quad$ List of Questions

  [TURN OVER

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• Paper 4, Section II, J

  #Optimization and Control

  2014

  A girl begins swimming from a point $(0,0)$ on the bank of a straight river. She swims at a constant speed $v$ relative to the water. The speed of the downstream current at a distance $y$ from the shore is $c(y)$. Hence her trajectory is described by

  $$\dot{x}=v \cos \theta+c(y), \quad \dot{y}=v \sin \theta$$

  where $\theta$ is the angle at which she swims relative to the direction of the current.

  She desires to reach a downstream point $(1,0)$ on the same bank as she starts, as quickly as possible. Construct the Hamiltonian for this problem, and describe how Pontryagin's maximum principle can be used to give necessary conditions that must hold on an optimal trajectory. Given that $c(y)$ is positive, increasing and differentiable in $y$, show that on an optimal trajectory

  $$\frac{d}{d t} \tan (\theta(t))=-c^{\prime}(y(t))$$

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• Paper 3, Section II, J

  #Optimization and Control

  2014

  A particle follows a discrete-time trajectory on $\mathbb{R}$ given by

  $$x_{t+1}=A x_{t}+\xi_{t} u_{t}+\epsilon_{t}$$

  for $t=1,2, \ldots, T$, where $T \geqslant 2$ is a fixed integer, $A$ is a real constant, $x_{t}$ is the position of the particle and $u_{t}$ is the control action at time $t$, and $\left(\xi_{t}, \epsilon_{t}\right)_{t=1}^{T}$ is a sequence of independent random vectors with $\mathbb{E} \xi_{t}=\mathbb{E} \epsilon_{t}=0, \operatorname{var}\left(\xi_{t}\right)=V_{\xi}>0, \operatorname{var}\left(\epsilon_{t}\right)=V_{\epsilon}>0$ and $\operatorname{cov}\left(\xi_{t}, \epsilon_{t}\right)=0$.

  Find the closed-loop control, i.e. the control action $u_{t}$ defined as a function of $\left(x_{1}, \ldots, x_{t} ; u_{1}, \ldots, u_{t-1}\right)$, that minimizes

  $$\sum_{t=1}^{T} x_{t}^{2}+c \sum_{t=1}^{T-1} u_{t}$$

  where $c>0$ is given. [Note that this function is quadratic in $x$, but linear in $u$.]

  Does the closed-loop control depend on $V_{\epsilon}$ or on $V_{\xi}$ ? Deduce the form of the optimal open-loop control.

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• Paper 2, Section II, J

  #Optimization and Control

  2014

  Describe the elements of a discrete-time stochastic dynamic programming equation for the problem of maximizing the expected sum of non-negative rewards over an infinite horizon. Give an example to show that there may not exist an optimal policy. Prove that if a policy has a value function that satisfies the dynamic programming equation then the policy is optimal.

  A squirrel collects nuts for the coming winter. There are plenty of nuts lying around, but each time the squirrel leaves its lair it risks being caught by a predator. Assume that the outcomes of the squirrel's journeys are independent, that it is caught with probability $p$, and that it returns safely with a random weight of nuts, exponentially distributed with parameter $\lambda$. By solving the dynamic programming equation for the value function $F(x)$, find a policy maximizing the expected weight of nuts collected for the winter. Here the state variable $x$ takes values in $\mathbb{R}_{+}$(the weight of nuts so far collected) or $-1$ (a no-return state when the squirrel is caught).

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• Paper 4, Section II, D

  #Partial Differential Equations

  2014

  (a) Derive the solution of the one-dimensional wave equation

  $$u_{t t}-u_{x x}=0, \quad u(0, x)=u_{0}(x), \quad u_{t}(0, x)=u_{1}(x),$$

  with Cauchy data given by $C^{2}$ functions $u_{j}=u_{j}(x), j=0,1$, and where $x \in \mathbb{R}$ and $u_{t t}=\partial_{t}^{2} u$ etc. Explain what is meant by the property of finite propagation speed for the wave equation. Verify that the solution to (1) satisfies this property.

  (b) Consider the Cauchy problem

  $$u_{t t}-u_{x x}+x^{2} u=0, \quad u(0, x)=u_{0}(x), \quad u_{t}(0, x)=u_{1}(x)$$

  By considering the quantities

  $$e=\frac{1}{2}\left(u_{t}^{2}+u_{x}^{2}+x^{2} u^{2}\right) \quad \text { and } \quad p=-u_{t} u_{x}$$

  prove that solutions of (2) also satisfy the property of finite propagation speed.

  (c) Define what is meant by a strongly continuous one-parameter group of unitary operators on a Hilbert space. Consider the Cauchy problem for the Schrödinger equation for $\psi(x, t) \in \mathbb{C}$ :

  $$i \psi_{t}=-\psi_{x x}+x^{2} \psi, \quad \psi(x, 0)=\psi_{0}(x), \quad-\infty<x<\infty$$

  [In the following you may use without proof the fact that there is an orthonormal set of (real-valued) Schwartz functions $\left\{f_{j}(x)\right\}_{j=1}^{\infty}$ which are eigenfunctions of the differential operator $P=-\partial_{x}^{2}+x^{2}$ with eigenvalues $2 j+1$, i.e.

  $$P f_{j}=(2 j+1) f_{j}, \quad f_{j} \in \mathcal{S}(\mathbb{R}), \quad\left(f_{j}, f_{k}\right)_{L^{2}}=\int_{\mathbb{R}} f_{j}(x) f_{k}(x) d x=\delta_{j k},$$

  and which have the property that any function $u \in L^{2}$ can be written uniquely as a sum $u(x)=\sum_{j}\left(f_{j}, u\right)_{L^{2}} f_{j}(x)$ which converges in the metric defined by the $L^{2}$ norm.]

  Write down the solution to (3) in the case that $\psi_{0}$ is given by a finite sum $\psi_{0}=\sum_{j=1}^{N}\left(f_{j}, \psi_{0}\right)_{L^{2}} f_{j}$ and show that your formula extends to define a strongly continuous one-parameter group of unitary operators on the Hilbert space $L^{2}$ of square-integrable (complex-valued) functions, with inner product $(f, g)_{L^{2}}=\int_{\mathbb{R}} \overline{f(x)} g(x) d x$.

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• Paper 3, Section II, D

  #Partial Differential Equations

  2014

  (a) Consider variable-coefficient operators of the form

  $$P u=-\sum_{j, k=1}^{n} a_{j k} \partial_{j} \partial_{k} u+\sum_{j=1}^{n} b_{j} \partial_{j} u+c u$$

  whose coefficients are defined on a bounded open set $\Omega \subset \mathbb{R}^{n}$ with smooth boundary $\partial \Omega$. Let $a_{j k}$ satisfy the condition of uniform ellipticity, namely

  $$m\|\xi\|^{2} \leqslant \sum_{j, k=1}^{n} a_{j k}(x) \xi_{j} \xi_{k} \leqslant M\|\xi\|^{2} \quad \text { for all } x \in \Omega \text { and } \xi \in \mathbb{R}^{n}$$

  for suitably chosen positive numbers $m, M$.

  State and prove the weak maximum principle for solutions of $P u=0$. [Any results from linear algebra and calculus needed in your proof should be stated clearly, but need not be proved.]

  (b) Consider the nonlinear elliptic equation

  $$-\Delta u+e^{u}=f$$

  for $u: \mathbb{R}^{n} \rightarrow \mathbb{R}$ satisfying the additional condition

  $$\lim _{|x| \rightarrow \infty} u(x)=0 .$$

  Assume that $f \in \mathcal{S}\left(\mathbb{R}^{n}\right)$. Prove that any two $C^{2}$ solutions of (1) which also satisfy (2) are equal.

  Now let $u \in C^{2}\left(\mathbb{R}^{n}\right)$ be a solution of $(1)$ and (2). Prove that if $f(x)<1$ for all $x$ then $u(x)<0$ for all $x$. Prove that if $\max _{x} f(x)=L \geqslant 1$ then $u(x) \leqslant \ln L$ for all $x$.

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• Paper 1, Section II, D

  #Partial Differential Equations

  2014

  State the Cauchy-Kovalevskaya theorem, including a definition of the term noncharacteristic.

  For which values of the real number $a$, and for which functions $f$, does the CauchyKovalevskaya theorem ensure that the Cauchy problem

  $$u_{t t}=u_{x x}+a u_{x x x x}, \quad u(x, 0)=0, u_{t}(x, 0)=f(x)$$

  has a local solution?

  Now consider the Cauchy problem (1) in the case that $f(x)=\sum_{m \in \mathbb{Z}} \hat{f}(m) e^{i m x}$ is a smooth $2 \pi$-periodic function.

  (i) Show that if $a \leqslant 0$ there exists a unique smooth solution $u$ for all times, and show that for all $T \geqslant 0$ there exists a number $C=C(T)>0$, independent of $f$, such that

  $$\int_{-\pi}^{+\pi}|u(x, t)|^{2} d x \leqslant C \int_{-\pi}^{+\pi}|f(x)|^{2} d x$$

  for all $t:|t| \leqslant T$.

  (ii) If $a=1$ does there exist a choice of $C=C(T)$ for which (2) holds? Give a full justification for your answer.

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• Paper 2, Section II, D

  #Partial Differential Equations

  2014

  In this question, functions are all real-valued, and

  $$H_{p e r}^{s}=\left\{u=\sum_{m \in \mathbb{Z}} \hat{u}(m) e^{i m x} \in L^{2}:\|u\|_{H^{s}}^{2}=\sum_{m \in \mathbb{Z}}\left(1+m^{2}\right)^{s}|\hat{u}(m)|^{2}<\infty\right\}$$

  are the Sobolev spaces of functions $2 \pi$-periodic in $x$, for $s=0,1,2, \ldots$

  State Parseval's theorem. For $s=0,1$ prove that the norm $\|u\|_{H^{s}}$ is equivalent to the norm ||$_{s}$ defined by

  $$\|u\|_{s}^{2}=\sum_{r=0}^{s} \int_{-\pi}^{+\pi}\left(\partial_{x}^{r} u\right)^{2} d x$$

  Consider the Cauchy problem

  $$u_{t}-u_{x x}=f, \quad u(x, 0)=u_{0}(x), \quad t \geqslant 0,$$

  where $f=f(x, t)$ is a smooth function which is $2 \pi$-periodic in $x$, and the initial value $u_{0}$ is also smooth and $2 \pi$-periodic. Prove that if $u$ is a smooth solution which is $2 \pi$-periodic in $x$, then it satisfies

  $$\int_{0}^{T}\left(u_{t}^{2}+u_{x x}^{2}\right) d t \leqslant C\left(\left\|u_{0}\right\|_{H^{1}}^{2}+\int_{0}^{T} \int_{-\pi}^{\pi}|f(x, t)|^{2} d x d t\right)$$

  for some number $C>0$ which does not depend on $u$ or $f$.

  State the Lax-Milgram lemma. Prove, using the Lax-Milgram lemma, that if

  $$f(x, t)=e^{\lambda t} g(x)$$

  with $g \in H_{p e r}^{0}$ and $\lambda>0$, then there exists a weak solution to (1) of the form $u(x, t)=e^{\lambda t} \phi(x)$ with $\phi \in H_{\text {per. }}^{1}$. Does the same hold for all $\lambda \in \mathbb{R}$ ? Briefly explain your answer.

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• Paper 4, Section II, A

  #Principles of Quantum Mechanics

  2014

  Define the interaction picture for a quantum mechanical system with Schrödinger picture Hamiltonian $H_{0}+V(t)$ and explain why the interaction and Schrödinger pictures give the same physical predictions for transition rates between eigenstates of $H_{0}$. Derive the equation of motion for the interaction picture states $|\overline{\psi(t)}\rangle$.

  A system consists of just two states $|1\rangle$ and $|2\rangle$, with respect to which

  $$H_{0}=\left(\begin{array}{cc} E_{1} & 0 \\ 0 & E_{2} \end{array}\right), \quad V(t)=\hbar \lambda\left(\begin{array}{cc} 0 & e^{i \omega t} \\ e^{-i \omega t} & 0 \end{array}\right)$$

  Writing the interaction picture state as $\overline{\langle(t)}\rangle=a_{1}(t)|1\rangle+a_{2}(t)|2\rangle$, show that the interaction picture equation of motion can be written as

  $$i \dot{a}_{1}(t)=\lambda e^{i \mu t} a_{2}(t), \quad i \dot{a}_{2}(t)=\lambda e^{-i \mu t} a_{1}(t)$$

  where $\mu=\omega-\omega_{21}$ and $\omega_{21}=\left(E_{2}-E_{1}\right) / \hbar$. Hence show that $a_{2}(t)$ satisfies

  $$\ddot{a}_{2}+i \mu \dot{a}_{2}+\lambda^{2} a_{2}=0 .$$

  Given that $a_{2}(0)=0$, show that the solution takes the form

  $$a_{2}(t)=\alpha e^{-i \mu t / 2} \sin \Omega t,$$

  where $\Omega$ is a frequency to be determined and $\alpha$ is a complex constant of integration.

  Substitute this solution for $a_{2}(t)$ into $(*)$ to determine $a_{1}(t)$ and, by imposing the normalization condition $\||\overline{\psi(t)}\rangle \|^{2}=1$ at $t=0$, show that $|\alpha|^{2}=\lambda^{2} / \Omega^{2}$.

  At time $t=0$ the system is in the state $|1\rangle$. Write down the probability of finding the system in the state $|2\rangle$ at time $t$.

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• Paper 3, Section II, A

  #Principles of Quantum Mechanics

  2014

  Let $\mathbf{J}=\left(J_{1}, J_{2}, J_{3}\right)$ and $|j m\rangle$ denote the standard angular-momentum operators and states so that, in units where $\hbar=1$,

  $$\mathbf{J}^{2}|j m\rangle=j(j+1)|j m\rangle, \quad J_{3}|j m\rangle=m|j m\rangle$$

  Show that $U(\theta)=\exp \left(-i \theta J_{2}\right)$ is unitary. Define

  $$J_{i}(\theta)=U(\theta) J_{i} U^{-1}(\theta) \quad \text { for } i=1,2,3$$

  and

  $$|j m\rangle_{\theta}=U(\theta)|j m\rangle \text {. }$$

  Find expressions for $J_{1}(\theta), J_{2}(\theta)$ and $J_{3}(\theta)$ as linear combinations of $J_{1}, J_{2}$ and $J_{3}$. Briefly explain why $U(\theta)$ represents a rotation of $\mathbf{J}$ through angle $\theta$ about the 2-axis.

  Show that

  $$J_{3}(\theta)|j m\rangle_{\theta}=m|j m\rangle_{\theta}$$

  Express $|10\rangle_{\theta}$ as a linear combination of the states $|1 m\rangle, m=-1,0,1$. By expressing $J_{1}$ in terms of $J_{\pm}$, use $(*)$ to determine the coefficients in this expansion.

  A particle of spin 1 is in the state $|10\rangle$ at time $t=0$. It is subject to the Hamiltonian

  $$H=-\mu \mathbf{B} \cdot \mathbf{J}$$

  where $\mathbf{B}=(0, \mathbf{B}, 0)$. At time $t$ the value of $J_{3}$ is measured and found to be $J_{3}=0$. At time $2 t$ the value of $J_{3}$ is measured again and found to be $J_{3}=1$. Show that the joint probability for these two values to be measured is

  $$\frac{1}{8} \sin ^{2}(2 \mu B t) .$$

  [The following result may be quoted: $\left.J_{\pm}|j m\rangle=\sqrt{(j \mp m)(j \pm m+1)}|j m \pm 1\rangle .\right]$

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• Paper 2, Section II, A

  #Principles of Quantum Mechanics

  2014

  (i) Let $a$ and $a^{\dagger}$ be the annihilation and creation operators, respectively, for a simple harmonic oscillator whose Hamiltonian is

  $$H_{0}=\omega\left(a^{\dagger} a+\frac{1}{2}\right),$$

  with $\left[a, a^{\dagger}\right]=1$. Explain how the set of eigenstates $\{|n\rangle: n=0,1,2, \ldots\}$ of $H_{0}$ is obtained and deduce the corresponding eigenvalues. Show that

  (ii) Consider a system whose unperturbed Hamiltonian is

  $$H_{0}=\left(a^{\dagger} a+\frac{1}{2}\right)+2\left(b^{\dagger} b+\frac{1}{2}\right)$$

  where $\left[a, a^{\dagger}\right]=1, \quad\left[b, b^{\dagger}\right]=1$ and all other commutators are zero. Find the degeneracies of the eigenvalues of $H_{0}$ with energies $E_{0}=\frac{3}{2}, \frac{5}{2}, \frac{7}{2}, \frac{9}{2}$ and $\frac{11}{2}$.

  The system is perturbed so that it is now described by the Hamiltonian

  $$H=H_{0}+\lambda H^{\prime},$$

  where $H^{\prime}=\left(a^{\dagger}\right)^{2} b+a^{2} b^{\dagger}$. Using degenerate perturbation theory, calculate to $O(\lambda)$ the energies of the eigenstates associated with the level $E_{0}=\frac{9}{2}$.

  Write down the eigenstates, to $O(\lambda)$, associated with these perturbed energies. By explicit evaluation show that they are in fact exact eigenstates of $H$ with these energies as eigenvalues.

  $$\begin{aligned} & a|0\rangle=0 \\ & a|n\rangle=\sqrt{n}|n-1\rangle, \quad n \geqslant 1, \\ & a^{\dagger}|n\rangle=\sqrt{n+1}|n+1\rangle, \quad n \geqslant 0 . \end{aligned}$$

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• Paper 1, Section II, A

  #Principles of Quantum Mechanics

  2014

  Let $\hat{x}, \hat{p}$ and $H(\hat{x}, \hat{p})=\hat{p}^{2} / 2 m+V(\hat{x})$ be the position operator, momentum operator and Hamiltonian for a particle moving in one dimension. Let $|\psi\rangle$ be the state vector for the particle. The position and momentum eigenstates have inner products

  $$\langle x \mid p\rangle=\frac{1}{\sqrt{2 \pi \hbar}} \exp (i p x / \hbar), \quad\left\langle x \mid x^{\prime}\right\rangle=\delta\left(x-x^{\prime}\right) \quad \text { and } \quad\left\langle p \mid p^{\prime}\right\rangle=\delta\left(p-p^{\prime}\right) .$$

  Show that

  $$\langle x|\hat{p}| \psi\rangle=-i \hbar \frac{\partial}{\partial x} \psi(x) \quad \text { and } \quad\langle p|\hat{x}| \psi\rangle=i \hbar \frac{\partial}{\partial p} \tilde{\psi}(p)$$

  where $\psi(x)$ and $\tilde{\psi}(p)$ are the wavefunctions in the position representation and momentum representation, respectively. Show how $\psi(x)$ and $\tilde{\psi}(p)$ may be expressed in terms of each other.

  For general $V(\hat{x})$, express $\langle p|V(\hat{x})| \psi\rangle$ in terms of $\tilde{\psi}(p)$, and hence write down the time-independent Schrödinger equation in the momentum representation satisfied by $\tilde{\psi}(p)$.

  Consider now the case $V(x)=-\left(\hbar^{2} \lambda / m\right) \delta(x), \lambda>0$. Show that there is a bound state with energy $E=-\varepsilon, \varepsilon>0$, with wavefunction $\tilde{\psi}(p)$ satisfying

  $$\tilde{\psi}(p)=\frac{\hbar \lambda}{\pi} \frac{1}{2 m \varepsilon+p^{2}} \int_{-\infty}^{\infty} \tilde{\psi}\left(p^{\prime}\right) d p^{\prime}$$

  Hence show that there is a unique value for $\varepsilon$ and determine this value.

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• Paper 4, Section II, J

  #Principles of Statistics

  2014

  Suppose you have at hand a pseudo-random number generator that can simulate an i.i.d. sequence of uniform $U[0,1]$ distributed random variables $U_{1}^{*}, \ldots, U_{N}^{*}$ for any $N \in \mathbb{N}$. Construct an algorithm to simulate an i.i.d. sequence $X_{1}^{*}, \ldots, X_{N}^{*}$ of standard normal $N(0,1)$ random variables. [Should your algorithm depend on the inverse of any cumulative probability distribution function, you are required to provide an explicit expression for this inverse function.]

  Suppose as a matter of urgency you need to approximately evaluate the integral

  $$I=\frac{1}{\sqrt{2 \pi}} \int_{\mathbb{R}} \frac{1}{(\pi+|x|)^{1 / 4}} e^{-x^{2} / 2} d x$$

  Find an approximation $I_{N}$ of this integral that requires $N$ simulation steps from your pseudo-random number generator, and which has stochastic accuracy

  $$\operatorname{Pr}\left(\left|I_{N}-I\right|>N^{-1 / 4}\right) \leqslant N^{-1 / 2}$$

  where Pr denotes the joint law of the simulated random variables. Justify your answer.

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• Paper 3, Section II, J

  #Principles of Statistics

  2014

  State and prove Wilks' theorem about testing the simple hypothesis $H_{0}: \theta=\theta_{0}$, against the alternative $H_{1}: \theta \in \Theta \backslash\left\{\theta_{0}\right\}$, in a one-dimensional regular parametric model $\{f(\cdot, \theta): \theta \in \Theta\}, \Theta \subseteq \mathbb{R}$. [You may use without proof the results from lectures on the consistency and asymptotic distribution of maximum likelihood estimators, as well as on uniform laws of large numbers. Necessary regularity conditions can be assumed without statement.]

  Find the maximum likelihood estimator $\hat{\theta}_{n}$ based on i.i.d. observations $X_{1}, \ldots, X_{n}$ in a $N(0, \theta)$-model, $\theta \in \Theta=(0, \infty)$. Deduce the limit distribution as $n \rightarrow \infty$ of the sequence of statistics

  $$-n\left(\log \left(\overline{X^{2}}\right)-\left(\overline{X^{2}}-1\right)\right)$$

  where $\overline{X^{2}}=(1 / n) \sum_{i=1}^{n} X_{i}^{2}$ and $X_{1}, \ldots, X_{n}$ are i.i.d. $N(0,1)$.

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• Paper 2, Section II, J

  #Principles of Statistics

  2014

  In a general decision problem, define the concepts of a Bayes rule and of admissibility. Show that a unique Bayes rule is admissible.

  Consider i.i.d. observations $X_{1}, \ldots, X_{n}$ from a $\operatorname{Poisson}(\theta), \theta \in \Theta=(0, \infty)$, model. Can the maximum likelihood estimator $\hat{\theta}_{M L E}$ of $\theta$ be a Bayes rule for estimating $\theta$ in quadratic risk for any prior distribution on $\theta$ that has a continuous probability density on $(0, \infty) ?$ Justify your answer.

  Now model the $X_{i}$ as i.i.d. copies of $X \mid \theta \sim \operatorname{Poisson}(\theta)$, where $\theta$ is drawn from a prior that is a Gamma distribution with parameters $\alpha>0$ and $\lambda>0$ (given below). Show that the posterior distribution of $\theta \mid X_{1}, \ldots, X_{n}$ is a Gamma distribution and find its parameters. Find the Bayes rule $\hat{\theta}_{B A Y E S}$ for estimating $\theta$ in quadratic risk for this prior. [The Gamma probability density function with parameters $\alpha>0, \lambda>0$ is given by

  $$f(\theta)=\frac{\lambda^{\alpha} \theta^{\alpha-1} e^{-\lambda \theta}}{\Gamma(\alpha)}, \quad \theta>0$$

  where $\Gamma(\alpha)$ is the usual Gamma function.]

  Finally assume that the $X_{i}$ have actually been generated from a fixed Poisson $\left(\theta_{0}\right)$ distribution, where $\theta_{0}>0$. Show that $\sqrt{n}\left(\hat{\theta}_{B A Y E S}-\hat{\theta}_{M L E}\right)$ converges to zero in probability and deduce the asymptotic distribution of $\sqrt{n}\left(\hat{\theta}_{B A Y E S}-\theta_{0}\right)$ under the joint law $P_{\theta_{0}}^{\mathbb{N}}$ of the random variables $X_{1}, X_{2}, \ldots$. [You may use standard results from lectures without proof provided they are clearly stated.]

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• Paper 1, Section II, J

  #Principles of Statistics

  2014

  State without proof the inequality known as the Cramér-Rao lower bound in a parametric model $\{f(\cdot, \theta): \theta \in \Theta\}, \Theta \subseteq \mathbb{R}$. Give an example of a maximum likelihood estimator that attains this lower bound, and justify your answer.

  Give an example of a parametric model where the maximum likelihood estimator based on observations $X_{1}, \ldots, X_{n}$ is biased. State without proof an analogue of the Cramér-Rao inequality for biased estimators.

  Define the concept of a minimax decision rule, and show that the maximum likelihood estimator $\hat{\theta}_{M L E}$ based on $X_{1}, \ldots, X_{n}$ in a $N(\theta, 1)$ model is minimax for estimating $\theta \in \Theta=\mathbb{R}$ in quadratic risk.

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• Paper 4, Section II, K

  #Probability and Measure

  2014

  Let $\left(X_{n}: n \in \mathbb{N}\right)$ be a sequence of independent identically distributed random variables. Set $S_{n}=X_{1}+\cdots+X_{n}$.

  (i) State the strong law of large numbers in terms of the random variables $X_{n}$.

  (ii) Assume now that the $X_{n}$ are non-negative and that their expectation is infinite. Let $R \in(0, \infty)$. What does the strong law of large numbers say about the limiting behaviour of $S_{n}^{R} / n$, where $S_{n}^{R}=\left(X_{1} \wedge R\right)+\cdots+\left(X_{n} \wedge R\right)$ ?

  Deduce that $S_{n} / n \rightarrow \infty$ almost surely.

  Show that

  $$\sum_{n=0}^{\infty} \mathbb{P}\left(X_{n} \geqslant n\right)=\infty$$

  Show that $X_{n} \geqslant R n$ infinitely often almost surely.

  (iii) Now drop the assumption that the $X_{n}$ are non-negative but continue to assume that $\mathbb{E}\left(\left|X_{1}\right|\right)=\infty$. Show that, almost surely,

  $$\limsup _{n \rightarrow \infty}\left|S_{n}\right| / n=\infty$$

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• Paper 3, Section II, K

  #Probability and Measure

  2014

  (i) Let $(E, \mathcal{E}, \mu)$ be a measure space. What does it mean to say that a function $\theta: E \rightarrow E$ is a measure-preserving transformation?

  What does it mean to say that $\theta$ is ergodic?

  State Birkhoff's almost everywhere ergodic theorem.

  (ii) Consider the set $E=(0,1]^{2}$ equipped with its Borel $\sigma$-algebra and Lebesgue measure. Fix an irrational number $a \in(0,1]$ and define $\theta: E \rightarrow E$ by

  $$\theta\left(x_{1}, x_{2}\right)=\left(x_{1}+a, x_{2}+a\right)$$

  where addition in each coordinate is understood to be modulo 1 . Show that $\theta$ is a measurepreserving transformation. Is $\theta$ ergodic? Justify your answer.

  Let $f$ be an integrable function on $E$ and let $\bar{f}$ be the invariant function associated with $f$ by Birkhoff's theorem. Write down a formula for $\bar{f}$ in terms of $f$. [You are not expected to justify this answer.]

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• Paper 2, Section II, $26 K$

  #Probability and Measure

  2014

  State and prove the monotone convergence theorem.

  Let $\left(E_{1}, \mathcal{E}_{1}, \mu_{1}\right)$ and $\left(E_{2}, \mathcal{E}_{2}, \mu_{2}\right)$ be finite measure spaces. Define the product $\sigma$-algebra $\mathcal{E}=\mathcal{E}_{1} \otimes \mathcal{E}_{2}$ on $E_{1} \times E_{2}$.

  Define the product measure $\mu=\mu_{1} \otimes \mu_{2}$ on $\mathcal{E}$, and show carefully that $\mu$ is countably additive.

  [You may use without proof any standard facts concerning measurability provided these are clearly stated.]

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• Paper 1, Section II, $26 \mathrm{~K}$

  #Probability and Measure

  2014

  What is meant by the Borel $\sigma$-algebra on the real line $\mathbb{R}$ ?

  Define the Lebesgue measure of a Borel subset of $\mathbb{R}$ using the concept of outer measure.

  Let $\mu$ be the Lebesgue measure on $\mathbb{R}$. Show that, for any Borel set $B$ which is contained in the interval $[0,1]$, and for any $\varepsilon>0$, there exist $n \in \mathbb{N}$ and disjoint intervals $I_{1}, \ldots, I_{n}$ contained in $[0,1]$ such that, for $A=I_{1} \cup \cdots \cup I_{n}$, we have

  $$\mu(A \triangle B) \leqslant \varepsilon$$

  where $A \triangle B$ denotes the symmetric difference $(A \backslash B) \cup(B \backslash A)$.

  Show that there does not exist a Borel set $B$ contained in $[0,1]$ such that, for all intervals $I$ contained in $[0,1]$,

  $$\mu(B \cap I)=\mu(I) / 2$$

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• Paper 4, Section II, H

  #Representation Theory

  2014

  Let $G=\mathrm{SU}(2)$.

  (i) Sketch a proof that there is an isomorphism of topological groups $G /\{\pm I\} \cong$ $\mathrm{SO}(3)$

  (ii) Let $V_{2}$ be the irreducible complex representation of $G$ of dimension 3. Compute the character of the (symmetric power) representation $S^{n}\left(V_{2}\right)$ of $G$ for any $n \geqslant 0$. Show that the dimension of the space of invariants $\left(S^{n}\left(V_{2}\right)\right)^{G}$, meaning the subspace of $S^{n}\left(V_{2}\right)$ where $G$ acts trivially, is 1 for $n$ even and 0 for $n$ odd. [Hint: You may find it helpful to restrict to the unit circle subgroup $S^{1} \leqslant G$. The irreducible characters of $G$ may be quoted without proof.]

  Using the fact that $V_{2}$ yields the standard 3-dimensional representation of $\mathrm{SO}(3)$, show that $\bigoplus_{n \geqslant 0} S^{n} V_{2} \cong \mathbb{C}[x, y, z]$. Deduce that the ring of complex polynomials in three variables $x, y, z$ which are invariant under the action of $\mathrm{SO}(3)$ is a polynomial ring in one generator. Find a generator for this polynomial ring.

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• Paper 3, Section II, H

  #Representation Theory

  2014

  (i) State Frobenius' theorem for transitive permutation groups acting on a finite set. Define Frobenius group and show that any finite Frobenius group (with an appropriate action) satisfies the hypotheses of Frobenius' theorem.

  (ii) Consider the group

  $$F_{p, q}:=\left\langle a, b: a^{p}=b^{q}=1, b^{-1} a b=a^{u}\right\rangle$$

  where $p$ is prime, $q$ divides $p-1$ ( $q$ not necessarily prime), and $u$ has multiplicative order $q$ modulo $p$ (such elements $u$ exist since $q$ divides $p-1)$. Let $S$ be the subgroup of $\mathbb{Z}_{p}^{\times}$ consisting of the powers of $u$, so that $|S|=q$. Write $r=(p-1) / q$, and let $v_{1}, \ldots, v_{r}$ be coset representatives for $S$ in $\mathbb{Z}_{p}^{\times}$.

  (a) Show that $F_{p, q}$ has $q+r$ conjugacy classes and that a complete list of the classes comprises $\{1\},\left\{a^{v_{j} s}: s \in S\right\}(1 \leqslant j \leqslant r)$ and $\left\{a^{m} b^{n}: 0 \leqslant m \leqslant p-1\right\}(1 \leqslant n \leqslant q-1)$.

  (b) By observing that the derived subgroup $F_{p, q}^{\prime}=\langle a\rangle$, find $q$ 1-dimensional characters of $F_{p, q}$. [Appropriate results may be quoted without proof.]

  (c) Let $\varepsilon=e^{2 \pi i / p}$. For $v \in \mathbb{Z}_{p}^{\times}$denote by $\psi_{v}$ the character of $\langle a\rangle$ defined by $\psi_{v}\left(a^{x}\right)=\varepsilon^{v x}(0 \leqslant x \leqslant p-1)$. By inducing these characters to $F_{p, q}$, or otherwise, find $r$ distinct irreducible characters of degree $q$.

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• Paper 2, Section II, H

  #Representation Theory

  2014

  In this question work over $\mathbb{C}$. Let $H$ be a subgroup of $G$. State Mackey's restriction formula, defining all the terms you use. Deduce Mackey's irreducibility criterion.

  Let $G=\left\langle g, r: g^{m}=r^{2}=1, r g r^{-1}=g^{-1}\right\rangle$ (the dihedral group of order $2 m$ ) and let $H=\langle g\rangle$ (the cyclic subgroup of $G$ of order $m$ ). Write down the $m$ inequivalent irreducible characters $\chi_{k}(1 \leqslant k \leqslant m)$ of $H$. Determine the values of $k$ for which the induced character $\operatorname{Ind}_{H}^{G} \chi_{k}$ is irreducible.

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• Paper 1, Section II, H

  #Representation Theory

  2014

  (i) Let $K$ be any field and let $\lambda \in K$. Let $J_{\lambda, n}$ be the $n \times n$ Jordan block

  $$J_{\lambda, n}=\left(\begin{array}{ccccc} \lambda & 1 & 0 & \cdots & 0 \\ 0 & \ddots & \ddots & & \vdots \\ \vdots & & \ddots & \ddots & 0 \\ \vdots & & & \ddots & 1 \\ 0 & \cdots & \cdots & 0 & \lambda \end{array}\right)$$

  Compute $J_{\lambda, n}^{r}$ for each $r \geqslant 0$.

  (ii) Let $G$ be a cyclic group of order $N$, and let $K$ be an algebraically closed field of characteristic $p \geqslant 0$. Determine all the representations of $G$ on vector spaces over $K$, up to equivalence. Which are irreducible? Which do not split as a direct sum $W \oplus W^{\prime}$, with $W \neq 0$ and $W^{\prime} \neq 0 ?$

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• Paper 3, Section II, H

  #Riemann Surfaces

  2014

  State the Uniformization Theorem.

  Show that any domain of $\mathbb{C}$ whose complement has more than one point is uniformized by the unit disc $\Delta .$ [You may use the fact that for $\mathbb{C}_{\infty}$ the group of automorphisms consists of Möbius transformations, and for $\mathbb{C}$ it consists of maps of the form $z \mapsto a z+b$ with $a \in \mathbb{C}^{*}$ and $b \in \mathbb{C}$.

  Let $X$ be the torus $\mathbb{C} / \Lambda$, where $\Lambda$ is a lattice. Given $p \in X$, show that $X \backslash\{p\}$ is uniformized by the unit $\operatorname{disc} \Delta$.

  Is it true that a holomorphic map from $\mathbb{C}$ to a compact Riemann surface of genus two must be constant? Justify your answer.

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• Paper 2, Section II, H

  #Riemann Surfaces

  2014

  State and prove the Valency Theorem and define the degree of a non-constant holomorphic map between compact Riemann surfaces.

  Let $X$ be a compact Riemann surface of genus $g$ and $\pi: X \rightarrow \mathbb{C}_{\infty}$ a holomorphic map of degree two. Find the cardinality of the set $R$ of ramification points of $\pi$. Find also the cardinality of the set of branch points of $\pi$. [You may use standard results from lectures provided they are clearly stated.]

  Define $\sigma: X \rightarrow X$ as follows: if $p \in R$, then $\sigma(p)=p$; otherwise, $\sigma(p)=q$ where $q$ is the unique point such that $\pi(q)=\pi(p)$ and $p \neq q$. Show that $\sigma$ is a conformal equivalence with $\pi \sigma=\pi$ and $\sigma \sigma=$ id.

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• Paper 1, Section II, H

  #Riemann Surfaces

  2014

  If $X$ is a Riemann surface and $p: Y \rightarrow X$ is a covering map of topological spaces, show that there is a conformal structure on $Y$ such that $p: Y \rightarrow X$ is analytic.

  Let $f(z)$ be the complex polynomial $z^{5}-1$. Consider the subspace $R$ of $\mathbb{C}^{2}=\mathbb{C} \times \mathbb{C}$ given by the equation $w^{2}=f(z)$, where $(z, w)$ denotes coordinates in $\mathbb{C}^{2}$, and let $\pi: R \rightarrow \mathbb{C}$ be the restriction of the projection map onto the first factor. Show that $R$ has the structure of a Riemann surface which makes $\pi$ an analytic map. If $\tau$ denotes projection onto the second factor, show that $\tau$ is also analytic. [You may assume that $R$ is connected.]

  Find the ramification points and the branch points of both $\pi$ and $\tau$. Compute also the ramification indices at the ramification points.

  Assuming that it is possible to add a point $P$ to $R$ so that $X=R \cup\{P\}$ is a compact Riemann surface and $\tau$ extends to a holomorphic map $\tau: X \rightarrow \mathbb{C}_{\infty}$ such that $\tau^{-1}(\infty)=\{P\}$, compute the genus of $X .$

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• Paper 4, Section I, $5 K$

  #Statistical Modelling

  2014

  Consider the normal linear model where the $n$-vector of responses $Y$ satisfies $Y=X \beta+\varepsilon$ with $\varepsilon \sim N_{n}\left(0, \sigma^{2} I\right)$ and $X$ is an $n \times p$ design matrix with full column rank. Write down a $(1-\alpha)$-level confidence set for $\beta$.

  Define the Cook's distance for the observation $\left(Y_{i}, x_{i}\right)$ where $x_{i}^{T}$ is the $i$ th row of $X$, and give its interpretation in terms of confidence sets for $\beta$.

  In the model above with $n=100$ and $p=4$, you observe that one observation has Cook's distance 3.1. Would you be concerned about the influence of this observation? Justify your answer.

  [Hint: You may find some of the following facts useful:

  1. If $Z \sim \chi_{4}^{2}$, then $\mathbb{P}(Z \leqslant 1.06)=0.1, \mathbb{P}(Z \leqslant 7.78)=0.9$.

  2. If $Z \sim F_{4,96}$, then $\mathbb{P}(Z \leqslant 0.26)=0.1, \mathbb{P}(Z \leqslant 2.00)=0.9$.

  3. If $Z \sim F_{96,4}$, then $\left.\mathbb{P}(Z \leqslant 0.50)=0.1, \mathbb{P}(Z \leqslant 3.78)=0.9 .\right]$

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• Paper 3, Section I, $5 K$

  #Statistical Modelling

  2014

  In an experiment to study factors affecting the production of the plastic polyvinyl chloride $(\mathrm{PVC})$, three experimenters each used eight devices to produce the PVC and measured the sizes of the particles produced. For each of the 24 combinations of device and experimenter, two size measurements were obtained.

  The experimenters and devices used for each of the 48 measurements are stored in $\mathrm{R}$ as factors in the objects experimenter and device respectively, with the measurements themselves stored in the vector psize. The following analysis was performed in $\mathrm{R}$.

  

  Let $X$ and $X_{0}$ denote the design matrices obtained by model.matrix(fit) and model.matrix (fit0) respectively, and let $Y$ denote the response psize. Let $P$ and $P_{0}$ denote orthogonal projections onto the column spaces of $X$ and $X_{0}$ respectively.

  For each of the following quantities, write down their numerical values if they appear in the analysis of variance table above; otherwise write 'unknown'.

  1. $\|(I-P) Y\|^{2}$

  2. $\left\|X\left(X^{T} X\right)^{-1} X^{T} Y\right\|^{2}$

  3. $\left\|\left(I-P_{0}\right) Y\right\|^{2}-\|(I-P) Y\|^{2}$

  4. $\frac{\left\|\left(P-P_{0}\right) Y\right\|^{2} / 14}{\|(I-P) Y\|^{2} / 24}$

  5. $\sum_{i=1}^{48} Y_{i} / 48$

  Out of the two models that have been fitted, which appears to be the more appropriate for the data according to the analysis performed, and why?

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• Paper 2, Section I, K

  #Statistical Modelling

  2014

  Define the concept of an exponential dispersion family. Show that the family of scaled binomial distributions $\frac{1}{n} \operatorname{Bin}(n, p)$, with $p \in(0,1)$ and $n \in \mathbb{N}$, is of exponential dispersion family form.

  Deduce the mean of the scaled binomial distribution from the exponential dispersion family form.

  What is the canonical link function in this case?

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• Paper 1, Section $I$, K

  #Statistical Modelling

  2014

  Write down the model being fitted by the following $\mathrm{R}$ command, where $y \in\{0,1,2, \ldots\}^{n}$ and $X$ is an $n \times p$ matrix with real-valued entries.

  fit $<-\operatorname{glm}(\mathrm{y} \sim \mathrm{X}, \mathrm{family}=$ poisson)

  Write down the log-likelihood for the model. Explain why the command

  $\operatorname{sum}(y)-\operatorname{sum}($ predict (fit, type $=$ "response" $))$

  gives the answer 0, by arguing based on the log-likelihood you have written down. [Hint: Recall that if $Z \sim \operatorname{Pois}(\mu)$ then

  $$\mathbb{P}(Z=k)=\frac{\mu^{k} e^{-\mu}}{k !}$$

  for $k \in\{0,1,2, \ldots\}$.]

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• Paper 4, Section II, K

  #Statistical Modelling

  2014

  In a study on infant respiratory disease, data are collected on a sample of 2074 infants. The information collected includes whether or not each infant developed a respiratory disease in the first year of their life; the gender of each infant; and details on how they were fed as one of three categories (breast-fed, bottle-fed and supplement). The data are tabulated in $\mathrm{R}$ as follows:

  $\begin{array}{rrrrr} & \text { disease } & \text { nondisease } & \text { gender } & \text { food } \\ 1 & 77 & 381 & \text { Boy } & \text { Bottle-fed } \\ 2 & 19 & 128 & \text { Boy } & \text { Supplement } \\ 3 & 47 & 447 & \text { Boy } & \text { Breast-fed } \\ 4 & 48 & 336 & \text { Girl } & \text { Bottle-fed } \\ 5 & 16 & 111 & \text { Girl } & \text { Supplement } \\ 6 & 31 & 433 & \text { Girl } & \text { Breast-fed }\end{array}$

  Write down the model being fit by the $R$ commands on the following page:

  

  The following (slightly abbreviated) output from $R$ is obtained.

  

  Briefly explain the justification for the standard errors presented in the output above.

  Explain the relevance of the output of the following $R$ code to the data being studied, justifying your answer:

  $>\exp (c(-0.6693-1.96 * 0.153,-0.6693+1.96 * 0.153))$

  [1] $0.3793940 \quad 0.6911351$

  [Hint: It may help to recall that if $Z \sim N(0,1)$ then $\mathbb{P}(Z \geqslant 1.96)=0.025 .]$

  Let $D_{1}$ be the deviance of the model fitted by the following $\mathrm{R}$ command.

  $>$ fit $1<-$ glm (disease/total gender + food + gender:food,

  $+$ family = binomial, weights = total $)$

  What is the numerical value of $D_{1}$ ? Which of the two models that have been fitted should you prefer, and why?

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• Paper 1, Section II, $13 K$

  #Statistical Modelling

  2014

  Consider the normal linear model where the $n$-vector of responses $Y$ satisfies $Y=X \beta+\varepsilon$ with $\varepsilon \sim N_{n}\left(0, \sigma^{2} I\right)$. Here $X$ is an $n \times p$ matrix of predictors with full column rank where $n \geqslant p+3$, and $\beta \in \mathbb{R}^{p}$ is an unknown vector of regression coefficients. Let $X_{0}$ be the matrix formed from the first $p_{0}<p$ columns of $X$, and partition $\beta$ as $\beta=\left(\beta_{0}^{T}, \beta_{1}^{T}\right)^{T}$ where $\beta_{0} \in \mathbb{R}^{p_{0}}$ and $\beta_{1} \in \mathbb{R}^{p-p_{0}}$. Denote the orthogonal projections onto the column spaces of $X$ and $X_{0}$ by $P$ and $P_{0}$ respectively.

  It is desired to test the null hypothesis $H_{0}: \beta_{1}=0$ against the alternative hypothesis $H_{1}: \beta_{1} \neq 0$. Recall that the $F$-test for testing $H_{0}$ against $H_{1}$ rejects $H_{0}$ for large values of

  $$F=\frac{\left\|\left(P-P_{0}\right) Y\right\|^{2} /\left(p-p_{0}\right)}{\|(I-P) Y\|^{2} /(n-p)} .$$

  Show that $(I-P)\left(P-P_{0}\right)=0$, and hence prove that the numerator and denominator of $F$ are independent under either hypothesis.

  Show that

  $$\mathbb{E}_{\beta, \sigma^{2}}(F)=\frac{(n-p)\left(\tau^{2}+1\right)}{n-p-2}$$

  where $\tau^{2}=\frac{\left\|\left(P-P_{0}\right) X \beta\right\|^{2}}{\left(p-p_{0}\right) \sigma^{2}}$.

  [In this question you may use the following facts without proof: $P-P_{0}$ is an orthogonal projection with rank $p-p_{0}$; any $n \times n$ orthogonal projection matrix $\Pi$ satisfies $\|\Pi \varepsilon\|^{2} \sim \sigma^{2} \chi_{\nu}^{2}$, where $\nu=\operatorname{rank}(\Pi) ;$ and if $Z \sim \chi_{\nu}^{2}$ then $\mathbb{E}\left(Z^{-1}\right)=(\nu-2)^{-1}$ when $\left.\nu>2 .\right]$

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• Paper 4, Section II, E

  #Statistical Physics

  2014

  The Dieterici equation of state of a gas is

  $$P=\frac{k_{B} T}{v-b} \exp \left(-\frac{a}{k_{B} T v}\right)$$

  where $P$ is the pressure, $v=V / N$ is the volume divided by the number of particles, $T$ is the temperature, and $k_{B}$ is the Boltzmann constant. Provide a physical interpretation for the constants $a$ and $b$.

  Briefly explain how the Dieterici equation captures the liquid-gas phase transition. What is the maximum temperature at which such a phase transition can occur?

  The Gibbs free energy is given by

  $$G=E+P V-T S$$

  where $E$ is the energy and $S$ is the entropy. Explain why the Gibbs free energy is proportional to the number of particles in the system.

  On either side of a first-order phase transition the Gibbs free energies are equal. Use this fact to derive the Clausius-Clapeyron equation for a line along which there is a first-order liquid-gas phase transition,

  $$\frac{d P}{d T}=\frac{L}{T\left(V_{\text {gas }}-V_{\text {liquid }}\right)}$$

  where $L$ is the latent heat which you should define.

  Assume that the volume of liquid is negligible compared to the volume of gas and that the latent heat is constant. Further assume that the gas can be well approximated by the ideal gas law. Solve $(*)$ to obtain an equation for the phase-transition line in the $(P, T)$ plane.

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• Paper 3, Section II, E

  #Statistical Physics

  2014

  In the grand canonical ensemble, at temperature $T$ and chemical potential $\mu$, what is the probability of finding a system in a state with energy $E$ and particle number $N$ ?

  A particle with spin degeneracy $g_{s}$ and mass $m$ moves in $d \geqslant 2$ spatial dimensions with dispersion relation $E=\hbar^{2} k^{2} / 2 m$. Compute the density of states $g(E)$. [You may denote the area of a unit $(d-1)$-dimensional sphere as $S_{d-1}$.]

  Treating the particles as non-interacting fermions, determine the energy $E$ of a gas in terms of the pressure $P$ and volume $V$.

  Derive an expression for the Fermi energy in terms of the number density of particles. Compute the degeneracy pressure at zero temperature in terms of the number of particles and the Fermi energy.

  Show that at high temperatures the gas obeys the ideal gas law (up to small corrections which you need not compute).

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• Paper 2, Section II, E

  #Statistical Physics

  2014

  Briefly describe the microcanonical, canonical and grand canonical ensembles. Why do they agree in the thermodynamic limit?

  A harmonic oscillator in one spatial dimension has Hamiltonian

  $$H=\frac{p^{2}}{2 m}+\frac{m}{2} \omega^{2} x^{2}$$

  Here $p$ and $x$ are the momentum and position of the oscillator, $m$ is its mass and $\omega$ its frequency. The harmonic oscillator is placed in contact with a heat bath at temperature $T$. What is the relevant ensemble?

  Treating the harmonic oscillator classically, compute the mean energy $\langle E\rangle$, the energy fluctuation $\Delta E^{2}$ and the heat capacity $C$.

  Treating the harmonic oscillator quantum mechanically, compute the mean energy $\langle E\rangle$, the energy fluctuation $\Delta E^{2}$ and the heat capacity $C$.

  In what limit of temperature do the classical and quantum results agree? Explain why they differ away from this limit. Describe an experiment for which this difference has implications.

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• Paper 1, Section II, E

  #Statistical Physics

  2014

  Write down the equation of state and the internal energy of a monatomic ideal gas.

  Describe the meaning of an adiabatic process. Derive the equation for an adiabatic process in the pressure-volume $(P, V)$ plane for a monatomic ideal gas.

  Briefly describe the Carnot cycle. Sketch the Carnot cycle in the $(P, V)$ plane and in the temperature entropy $(T, S)$ plane.

  The Diesel cycle is an idealised version of the process realised in the Diesel engine. It consists of the following four reversible steps:

  

  Sketch the Diesel cycle for a monatomic gas in the $(P, V)$ plane and the $(T, S)$ plane. Determine the equations for the curves $B \rightarrow C$ and $D \rightarrow A$ in the $(T, S)$ plane.

  The efficiency $\eta$ of the cycle is defined as

  $$\eta=1-\frac{Q_{\text {out }}}{Q_{\text {in }}}$$

  where $Q_{\text {in }}$ is the heat entering the gas in step $B \rightarrow C$ and $Q_{\text {out }}$ is the heat leaving the gas in step $D \rightarrow A$. Calculate $\eta$ as a function of the temperatures at points $A, B, C$ and $D$.

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• Paper 4, Section II, K

  #Stochastic Financial Models

  2014

  Write down the Black-Scholes partial differential equation (PDE), and explain briefly its relevance to option pricing.

  Show how a change of variables reduces the Black-Scholes PDE to the heat equation:

  $$\begin{aligned} &\frac{\partial f}{\partial t}+\frac{1}{2} \frac{\partial^{2} f}{\partial x^{2}}=0 \text { for all }(t, x) \in[0, T) \times \mathbb{R}, \\ &f(T, x)=\varphi(x) \text { for all } x \in \mathbb{R} \end{aligned}$$

  where $\varphi$ is a given boundary function.

  Consider the following numerical scheme for solving the heat equation on the equally spaced grid $\left(t_{n}, x_{k}\right) \in[0, T] \times \mathbb{R}$ where $t_{n}=n \Delta t$ and $x_{k}=k \Delta x, n=0,1, \ldots, N$ and $k \in \mathbb{Z}$, and $\Delta t=T / N$. We approximate $f\left(t_{n}, x_{k}\right)$ by $f_{k}^{n}$ where

  $$0=\frac{f^{n+1}-f^{n}}{\Delta t}+\theta L f^{n+1}+(1-\theta) L f^{n}, \quad f_{k}^{N}=\varphi\left(x_{k}\right)$$

  and $\theta \in[0,1]$ is a constant and the operator $L$ is the matrix with non-zero entries $L_{k k}=-\frac{1}{(\Delta x)^{2}}$ and $L_{k, k+1}=L_{k, k-1}=\frac{1}{2(\Delta x)^{2}}$. By considering what happens when $\varphi(x)=\exp (i \omega x)$, show that the finite-difference scheme $(*)$ is stable if and only if

  $$1 \geqslant \lambda(2 \theta-1),$$

  where $\lambda \equiv \Delta t /(\Delta x)^{2}$. For what values of $\theta$ is the scheme $(*)$ unconditionally stable?

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• Paper 3, Section II, $29 K$

  #Stochastic Financial Models

  2014

  Derive the Black-Scholes formula $C\left(S_{0}, K, r, T, \sigma\right)$ for the time- 0 price of a European call option with expiry $T$ and strike $K$ written on an asset with volatility $\sigma$ and time-0 price $S_{0}$, and where $r$ is the riskless rate of interest. Explain what is meant by the delta hedge for this option, and determine it explicitly.

  In terms of the Black-Scholes call option price formula $C$, find the time- 0 price of a forward-starting option, which pays $\left(S_{T}-\lambda S_{t}\right)^{+}$at time $T$, where $0<t<T$ and $\lambda>0$ are given. Find the price of an option which pays $\max \left\{S_{T}, \lambda S_{t}\right\}$ at time $T$. How would this option be hedged?

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• Paper 1, Section II, K

  #Stochastic Financial Models

  2014

  Suppose that $\bar{S}_{t} \equiv\left(S_{t}^{0}, \ldots, S_{t}^{d}\right)^{T}$ denotes the vector of prices of $d+1$ assets at times $t=0,1, \ldots$, and that $\bar{\theta}_{t} \equiv\left(\theta_{t}^{0}, \ldots, \theta_{t}^{d}\right)^{T}$ denotes the vector of the numbers of the $d+1$ different assets held by an investor from time $t-1$ to time $t$. Assuming that asset 0 is a bank account paying zero interest, that is, $S_{t}^{0}=1$ for all $t \geqslant 0$, explain what is meant by the statement that the portfolio process $\left(\bar{\theta}_{t}\right)_{t \geqslant 0}$ is self-financing. If the portfolio process is self-financing, prove that for any $t>0$

  $$\bar{\theta}_{t} \cdot \bar{S}_{t}-\bar{\theta}_{0} \cdot \bar{S}_{0}=\sum_{j=1}^{t} \theta_{j} \cdot \Delta S_{j}$$

  where $S_{j} \equiv\left(S_{j}^{1}, \ldots, S_{j}^{d}\right)^{T}, \Delta S_{j}=S_{j}-S_{j-1}$, and $\theta_{j} \equiv\left(\theta_{j}^{1}, \ldots, \theta_{j}^{d}\right)^{T}$.

  Suppose now that the $\Delta S_{t}$ are independent with common $N(0, V)$ distribution. Let

  $$F(z)=\inf E\left[\sum_{t \geqslant 1}(1-\beta) \beta^{t}\left\{\left(\bar{\theta}_{t} \cdot \bar{S}_{t}-\bar{\theta}_{0} \cdot \bar{S}_{0}\right)^{2}+\sum_{j=1}^{t}\left|\Delta \theta_{j}\right|^{2}\right\} \mid \theta_{0}=z\right]$$

  where $\beta \in(0,1)$ and the infimum is taken over all self-financing portfolio processes $\left(\bar{\theta}_{t}\right)_{t \geqslant 0}$ with $\theta_{0}^{0}=0$. Explain why $F$ should satisfy the equation

  $$F(z)=\beta \inf _{y}\left[y \cdot V y+|y-z|^{2}+F(y)\right]$$

  If $Q$ is a positive-definite symmetric matrix satisfying the equation

  $$Q=\beta(V+I+Q)^{-1}(V+Q)$$

  show that $(*)$ has a solution of the form $F(z)=z \cdot Q z$.

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• Paper 2, Section II, K

  #Stochastic Financial Models

  2014

  An agent has expected-utility preferences over his possible wealth at time 1 ; that is, the wealth $Z$ is preferred to wealth $Z^{\prime}$ if and only if $E U(Z) \geqslant E U\left(Z^{\prime}\right)$, where the function $U: \mathbb{R} \rightarrow \mathbb{R}$ is strictly concave and twice continuously differentiable. The agent can trade in a market, with the time-1 value of his portfolio lying in an affine space $\mathcal{A}$ of random variables. Assuming cash can be held between time 0 and time 1 , define the agent's time-0 utility indifference price $\pi(Y)$ for a contingent claim with time-1 value $Y$. Assuming any regularity conditions you may require, prove that the map $Y \mapsto \pi(Y)$ is concave.

  

  Consider a market with two claims with time-1 values $X$ and $Y$. Their joint distribution is

  $$\left(\begin{array}{l} X \\ Y \end{array}\right) \sim N\left(\left(\begin{array}{l} \mu_{X} \\ \mu_{Y} \end{array}\right),\left(\begin{array}{ll} V_{X X} & V_{X Y} \\ V_{Y X} & V_{Y Y} \end{array}\right)\right) .$$

  At time 0 , arbitrary quantities of the claim $X$ can be bought at price $p$, but $Y$ is not marketed. Derive an explicit expression for $\pi(Y)$ in the case where

  $$U(x)=-\exp (-\gamma x)$$

  where $\gamma>0$ is a given constant.

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• Paper 4, Section I, $2 G$

  #Topics in Analysis

  2014

  State Liouville's theorem on approximation of algebraic numbers by rationals.

  Prove that the number $\sum_{n=0}^{\infty} \frac{1}{2^{n^{n}}}$ is transcendental.

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• Paper 3, Section I, G

  #Topics in Analysis

  2014

  State Runge's theorem about uniform approximation of holomorphic functions by polynomials.

  Let $\mathbb{R}_{+} \subset \mathbb{C}$ be the subset of non-negative real numbers and let

  $$\Delta=\{z \in \mathbb{C}:|z|<1\} .$$

  Prove that there is a sequence of complex polynomials which converges to the function $1 / z$ uniformly on each compact subset of $\Delta \backslash \mathbb{R}_{+}$.

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• Paper 2, Section I, G

  #Topics in Analysis

  2014

  State Chebyshev's equal ripple criterion.

  Let

  $$h(t)=\prod_{\ell=1}^{n}\left(t-\cos \frac{(2 \ell-1) \pi}{2 n}\right)$$

  Show that if $q(t)=\sum_{j=0}^{n} a_{j} t^{j}$ where $a_{0}, \ldots, a_{n}$ are real constants with $\left|a_{n}\right| \geqslant 1$, then

  $$\sup _{t \in[-1,1]}|h(t)| \leqslant \sup _{t \in[-1,1]}|q(t)|$$

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• Paper 1, Section I, $2 \mathrm{G}$

  #Topics in Analysis

  2014

  (i) State Brouwer's fixed point theorem in the plane and an equivalent theorem concerning mapping a triangle $T$ to its boundary $\partial T$.

  (ii) Let $A$ be a $3 \times 3$ matrix with positive real entries. Use the theorems you stated in (i) to prove that $A$ has an eigenvector with positive entries.

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• Paper 2, Section II, G

  #Topics in Analysis

  2014

  Let $\gamma:[0,1] \rightarrow \mathbb{C}$ be a continuous map never taking the value 0 and satisfying $\gamma(0)=\gamma(1)$. Define the degree (or winding number) $w(\gamma ; 0)$ of $\gamma$ about 0 . Prove the following:

  (i) $w(1 / \gamma ; 0)=w\left(\gamma^{-} ; 0\right)$, where $\gamma^{-}(t)=\gamma(1-t)$.

  (ii) If $\sigma:[0,1] \rightarrow \mathbb{C}$ is continuous, $\sigma(0)=\sigma(1)$ and $|\sigma(t)|<|\gamma(t)|$ for each $0 \leqslant t \leqslant 1$, then $w(\gamma+\sigma ; 0)=w(\gamma ; 0)$.

  (iii) If $\gamma_{m}:[0,1] \rightarrow \mathbb{C}, m=1,2, \ldots$, are continuous maps with $\gamma_{m}(0)=\gamma_{m}(1)$, which converge to $\gamma$ uniformly on $[0,1]$ as $m \rightarrow \infty$, then $w\left(\gamma_{m} ; 0\right)=w(\gamma ; 0)$ for sufficiently large $m$.

  Let $\alpha:[0,1] \rightarrow \mathbb{C} \backslash\{0\}$ be a continuous map such that $\alpha(0)=\alpha(1)$ and $\left|\alpha(t)-e^{2 \pi i t}\right| \leqslant 1$ for each $t \in[0,1]$. Deduce from the results of (ii) and (iii) that $w(\alpha ; 0)=1$.

  [You may not use homotopy invariance of the winding number without proof.]

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• Paper 3, Section II, G

  #Topics in Analysis

  2014

  Define what is meant by a nowhere dense set in a metric space. State a version of the Baire Category Theorem. Show that any complete non-empty metric space without isolated points is uncountable.

  Let $A$ be the set of real numbers whose decimal expansion does not use the digit 6 . (A terminating decimal representation is used when it exists.) Show that there exists a real number which cannot be written as $a+q$ with $a \in A$ and $q \in \mathbb{Q}$.

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• Paper 4, Section II, C

  #Waves

  2014

  A one-dimensional shock wave propagates at a constant speed along a tube aligned with the $x$-axis and containing a perfect gas. In the reference frame where the shock is at rest at $x=0$, the gas has speed $U_{0}$, density $\rho_{0}$ and pressure $p_{0}$ in the region $x<0$ and speed $U_{1}$, density $\rho_{1}$ and pressure $p_{1}$ in the region $x>0$.

  Write down equations of conservation of mass, momentum and energy across the shock. Show that

  $$\frac{\gamma}{\gamma-1}\left(\frac{p_{1}}{\rho_{1}}-\frac{p_{0}}{\rho_{0}}\right)=\frac{p_{1}-p_{0}}{2}\left(\frac{1}{\rho_{1}}+\frac{1}{\rho_{0}}\right)$$

  where $\gamma$ is the ratio of specific heats.

  From now on, assume $\gamma=2$ and let $P=p_{1} / p_{0}$. Show that $\frac{1}{3}<\rho_{1} / \rho_{0}<3$.

  The increase in entropy from $x<0$ to $x>0$ is given by $\Delta S=C_{V} \log \left(p_{1} \rho_{0}^{2} / p_{0} \rho_{1}^{2}\right)$, where $C_{V}$ is a positive constant. Show that $\Delta S$ is a monotonic function of $P$.

  If $\Delta S>0$, deduce that $P>1, \rho_{1} / \rho_{0}>1,\left(U_{0} / c_{0}\right)^{2}>1$ and $\left(U_{1} / c_{1}\right)^{2}<1$, where $c_{0}$ and $c_{1}$ are the sound speeds in $x<0$ and $x>0$, respectively. Given that $\Delta S$ must have the same sign as $U_{0}$ and $U_{1}$, interpret these inequalities physically in terms of the properties of the flow upstream and downstream of the shock.

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• Paper 2, Section II, 38C

  #Waves

  2014

  The function $\phi(x, t)$ satisfies the equation

  $$\frac{\partial^{2} \phi}{\partial t^{2}}-\frac{\partial^{2} \phi}{\partial x^{2}}=\frac{\partial^{4} \phi}{\partial x^{2} \partial t^{2}}$$

  Derive the dispersion relation, and sketch graphs of frequency, phase velocity and group velocity as functions of the wavenumber. In the case of a localised initial disturbance, will it be the shortest or the longest waves that are to be found at the front of a dispersing wave packet? Do the wave crests move faster or slower than the wave packet?

  Give the solution to the initial-value problem for which at $t=0$

  $$\phi=\int_{-\infty}^{\infty} A(k) e^{i k x} d k \quad \text { and } \quad \frac{\partial \phi}{\partial t}=0$$

  and $\phi(x, 0)$ is real. Use the method of stationary phase to obtain an approximation for $\phi(V t, t)$ for fixed $0<V<1$ and large $t$. If, in addition, $\phi(x, 0)=\phi(-x, 0)$, deduce an approximation for the sequence of times at which $\phi(V t, t)=0$.

  You are given that $\phi(t, t)$ decreases like $t^{-1 / 4}$ for large $t$. Give a brief physical explanation why this rate of decay is slower than for $0<V<1$. What can be said about $\phi(V t, t)$ for large $t$ if $V>1$ ? [Detailed calculation is not required in these cases.]

  [You may assume that $\int_{-\infty}^{\infty} e^{-a u^{2}} d u=\sqrt{\frac{\pi}{a}} \quad$ for $\left.\operatorname{Re}(a) \geqslant 0, a \neq 0 .\right]$

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• Paper 3, Section II, 39C

  #Waves

  2014

  The equations describing small-amplitude motions in a stably stratified, incompressible, inviscid fluid are

  $$\frac{\partial \tilde{\rho}}{\partial t}+w \frac{\mathrm{d} \rho_{0}}{\mathrm{~d} z}=0, \quad \rho_{0} \frac{\partial \mathbf{u}}{\partial t}=\tilde{\rho} \mathbf{g}-\nabla \tilde{p}, \quad \boldsymbol{\nabla} \cdot \mathbf{u}=0$$

  where $\rho_{0}(z)$ is the background stratification, $\tilde{\rho}(\mathbf{x}, t)$ and $\tilde{p}(\mathbf{x}, t)$ are the perturbations about an undisturbed hydrostatic state, $\mathbf{u}(\mathbf{x}, t)=(u, v, w)$ is the velocity, and $\mathbf{g}=(0,0,-g)$.

  Show that

  $$\left[\frac{\partial^{2}}{\partial t^{2}} \nabla^{2}+N^{2}\left(\nabla^{2}-\frac{\partial^{2}}{\partial z^{2}}\right)\right] w=0$$

  stating any approximation made, and define the Brunt-Väisälä frequency $N$.

  Deduce the dispersion relation for plane harmonic waves with wavevector $\mathbf{k}=$ $(k, 0, m)$. Calculate the group velocity and verify that it is perpendicular to $\mathbf{k}$.

  Such a stably stratified fluid with a uniform value of $N$ occupies the region $z>h(x, t)$ above a moving lower boundary $z=h(x, t)$. Find the velocity field $w(x, z, t)$ generated by the boundary motion for the case $h=\epsilon \sin [k(x-U t)]$, where $0<\epsilon k \ll 1$ and $U>0$ is a constant.

  For the case $k^{2}<N^{2} / U^{2}$, sketch the orientation of the wave crests, the direction of propagation of the crests, and the direction of the group velocity.

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• Paper 1, Section II, C

  #Waves

  2014

  State the equations that relate strain to displacement and stress to strain in a uniform, linear, isotropic elastic solid with Lamé moduli $\lambda$ and $\mu$. In the absence of body forces, the Cauchy momentum equation for the infinitesimal displacements $\mathbf{u}(\mathbf{x}, t)$ is

  $$\rho \frac{\partial^{2} \mathbf{u}}{\partial t^{2}}=\boldsymbol{\nabla} \cdot \boldsymbol{\sigma}$$

  where $\rho$ is the density and $\boldsymbol{\sigma}$ the stress tensor. Show that both the dilatation $\boldsymbol{\nabla} \cdot \mathbf{u}$ and the rotation $\nabla \wedge \mathbf{u}$ satisfy wave equations, and find the wave-speeds $c_{P}$ and $c_{S}$.

  A plane harmonic $\mathrm{P}$-wave with wavevector $\mathbf{k}$ lying in the $(x, z)$ plane is incident from $z<0$ at an oblique angle on the planar interface $z=0$ between two elastic solids with different densities and elastic moduli. Show in a diagram the directions of all the reflected and transmitted waves, labelled with their polarisations, assuming that none of these waves are evanescent. State the boundary conditions on components of $\mathbf{u}$ and $\boldsymbol{\sigma}$ that would, in principle, determine the amplitudes.

  Now consider a plane harmonic P-wave of unit amplitude incident with $\mathbf{k}=$ $k(\sin \theta, 0, \cos \theta)$ on the interface $z=0$ between two elastic (and inviscid) liquids with wave-speed $c_{P}$ and modulus $\lambda$ in $z<0$ and wave-speed $c_{P}^{\prime}$ and modulus $\lambda^{\prime}$ in $z>0$. Obtain solutions for the reflected and transmitted waves. Show that the amplitude of the reflected wave is zero if

  $$\sin ^{2} \theta=\frac{Z^{\prime 2}-Z^{2}}{Z^{\prime 2}-\left(c_{P}^{\prime} Z / c_{P}\right)^{2}}$$

  where $Z=\lambda / c_{P}$ and $Z^{\prime}=\lambda^{\prime} / c_{P}^{\prime}$

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• Paper 2, Section II, C

  #Variational Principles

  2014

  Write down the Euler-Lagrange equation for the integral

  $$\int f\left(y, y^{\prime}, x\right) d x$$

  An ant is walking on the surface of a sphere, which is parameterised by $\theta \in[0, \pi]($ angle from top of sphere) and $\phi \in[0,2 \pi$ ) (azimuthal angle). The sphere is sticky towards the top and the bottom and so the ant's speed is proportional to $\sin \theta$. Show that the ant's fastest route between two points will be of the form

  $$\sinh (A \phi+B)=\cot \theta$$

  for some constants $A$ and $B$. $[A, B$ need not be determined.]

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• Paper 4, Section II, C

  #Variational Principles

  2014

  Consider the integral

  $$I=\int f\left(y, y^{\prime}\right) d x$$

  Show that if $f$ satisfies the Euler-Lagrange equation, then

  $$f-y^{\prime} \frac{\partial f}{\partial y^{\prime}}=\text { constant. }$$

  An axisymmetric soap film $y(x)$ is formed between two circular wires at $x=\pm l$. The wires both have radius $r$. Show that the shape that minimises the surface area takes the form

  $$y(x)=k \cosh \frac{x}{k}$$

  Show that there exist two possible $k$ that satisfy the boundary conditions for $r / l$ sufficiently large.

  Show that for these solutions the second variation is given by

  $$\delta^{2} I=\pi \int_{-l}^{+l}\left(k \eta^{\prime 2}-\frac{1}{k} \eta^{2}\right) \operatorname{sech}^{2}\left(\frac{x}{k}\right) d x$$

  where $\eta$ is an axisymmetric perturbation with $\eta(\pm l)=0$.

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• Paper 1, Section II, C

  #Vectors and Matrices

  2014

  Let $A$ and $B$ be complex $n \times n$ matrices.

  (i) The commutator of $A$ and $B$ is defined to be

  $$[A, B] \equiv A B-B A$$

  Show that $[A, A]=0 ;[A, B]=-[B, A] ;$ and $[A, \lambda B]=\lambda[A, B]$ for $\lambda \in \mathbb{C}$. Show further that the trace of $[A, B]$ vanishes.

  (ii) A skew-Hermitian matrix $S$ is one which satisfies $S^{\dagger}=-S$, where $\dagger$ denotes the Hermitian conjugate. Show that if $A$ and $B$ are skew-Hermitian then so is $[A, B]$.

  (iii) Let $\mathcal{M}$ be the linear map from $\mathbb{R}^{3}$ to the set of $2 \times 2$ complex matrices given by

  $$\mathcal{M}\left(\begin{array}{l} x \\ y \\ z \end{array}\right)=x M_{1}+y M_{2}+z M_{3}$$

  where

  $$M_{1}=\frac{1}{2}\left(\begin{array}{cc} i & 0 \\ 0 & -i \end{array}\right), \quad M_{2}=\frac{1}{2}\left(\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}\right), \quad M_{3}=\frac{1}{2}\left(\begin{array}{cc} 0 & i \\ i & 0 \end{array}\right) \text {. }$$

  Prove that for any $\mathbf{a} \in \mathbb{R}^{3}, \mathcal{M}(\mathbf{a})$ is traceless and skew-Hermitian. By considering pairs such as $\left[M_{1}, M_{2}\right]$, or otherwise, show that for $\mathbf{a}, \mathbf{b} \in \mathbb{R}^{3}$,

  $$\mathcal{M}(\mathbf{a} \times \mathbf{b})=[\mathcal{M}(\mathbf{a}), \mathcal{M}(\mathbf{b})]$$

  (iv) Using the result of part (iii), or otherwise, prove that if $C$ is a traceless skewHermitian $2 \times 2$ matrix then there exist matrices $A, B$ such that $C=[A, B]$. [You may use geometrical properties of vectors in $\mathbb{R}^{3}$ without proof.]

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